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MATHEMATICAL  TEXTS  FOR  ENGINEERING  STUDENTS 
Edited  by  MANSFIELD  MERRIMAN 


MATllIvMATlCAL  TABLES 

FOR 

CLASS-EOOM  USE 


BY 

MANSFIELD  MEKRIMAN 

Editok-in-Chief  of  American  Civil  Engineers'  Pocket  Book 
Member  of  American  Mathematical  Society 


FIRST   EDITION 
FIRST    THOUSAND 


' 

!'?  !•  *  ' 

'  ,*  :''io,,i  i'  ''"' 

NEW   YORK 

JOHN   WILEY    & 

SONS, 

Inc. 

ndon:   chapman  & 

HALL, 

Limited 

1915 

7 


coptrioht,  191.5 
Bt 
MANSFIELD  MERRIMAN 


THE  SCIENTIFIC   PRESS 
ROBERT    DRUMMOND    »ND  COMPANY 

BROOKLYN,  N.  Y. 


PREFACE 


Computations  of  squares,  square  roots,  reciprocals,  areas 
of  circles,  and  other  functions  of  numbers  occur  in  the  daily  work 
of  every  student  of  technology.  This  book  gives  four-place 
tables  by  which  the  time  spent  in  such  computations  can  be  much 
shortened.  Tables  of  trigonometric  functions,  of  logarithms, 
and  of  weights  and  measures  are  also  given.  These  will  be  found 
useful  in  cases  where  a  high  degree  of  precision  is  not  required, 
namely,  in  the  great  majority  of  problems  that  arise  in  physics 
and  engineering. 

These  tables'have  been  taken  from  the  American  Civil  Engin- 
eers' Pocket  Book,  and  hence  have  a  form  which  is  approved  in 
the  work  of  practical  computation.  It  is  believed  that  they  will 
be  of  value  to  students  in  blackboard  work,  in  the  examination 
room,  in  the  field  and  laboratory,  and  in  the  drawing  room,  as  well 
as  in  the  solution  of  text-book  problems. 

Explanations,  prepared  especially  for  this  volume,  will  render 
the  use  of  the  tables  easily  understood,  and  these  are  accompanied 
by  remarks  regarding  the  degree  of  precision  to  be  expected  in 
interpolated  tabular  values.  Exercises  for  students  are  given  at 
the  end  of  each  chapter.  It  is  hoped  that  this  little  book  may 
assist  both  teachers  and  students  in  saving  time  and  labor,  and 
thus  tend  to  promote  economy  and  efficiency, 

3 


337855 


CONTENTS 


Chapter  1 
GENERAL   EXPLANATIONS 

PAGE 

Art.     1.  Arguments  and  Functions S 

2.  Precision  of  Tabular  Values 8 

3.  Interpolation 9 

4.  Precision  of  Computed  Results 10 

5.  Exercises  for  Students 10 

Chapter  2 
ARITHMETICAL  TABLES 

Art.     6.  Table  of  Reciprocals 12 

7.  Table  of  Squares 14 

8.  Table  of  Square  Roots 16 

9.  Table  of  Cubes 20 

10.  Table  of  Cube  Roots 22 

11.  Table  of  Three-Halves  Powers 23 

12.  Table  of  Fifth  Powers  and  Roots 24 

13.  Explanations  of  Tables 25 

14.  Exercises 26 

Chapter  3 
TABLES   OF   CIRCLES   AND   SPHERES 

Art.  15.  Table  of  Areas  of  Circles  (tenths)  .  .' 28 

16.  Table  of  Areas  of  Circles  (eighths) 30 

17.  Table  of  Circumferences  of  Circles  (tenths)     31 

18.  Table  of  Circumferences  of  Circles  (eighths)  ....  31 

19.  Table  of  Circular  Segments 32 

20.  Table  of  Volumes  of  Spheres  (tenths) 34 

21.  Table  of  Volumes  of  Spheres  (eighths) 34 

22.  Table  of  Multipliers  for  Circular  Arcs 35 

23.  Explanations 35 

24.  Exercises 36 

5 


6  Contents 

Chapter  4 
NATURAL   TRIGONOMETRIC   FUNCTIONS 

PAGE 

Art.  25.  Table  of  Sines  and  Cosines 38 

26.  Table  of  Tangents  and  Cotangents 40 

27.  Table  of  Four-Place  Trigonometric  Functions.  ...  42 

28.  Explanations 43 

29.  Exercises 44 

Chapter  5 
LOGARITHMIC    TABLES 

Art.  30.  Table  of  Common  Logarithms  of  Numbers 46 

31.  Table  of  Logarithms  of  Sines  and  Cosines 48 

32.  Table  of  Logarithms  of  Tangents  and  Cotangents  50 

33.  Table  of  Four-Place  Logarithms  of  Trigonometric 

Functions 52 

34.  Explanations 53 

35.  Exercises 54 

Chapter   6 
WEIGHTS   AND   MEASURES 

Art.  36.  Table  of  Measures  of  Length 56 

37.  Table  of  Measures  of  Area 56 

38.  Table  of  Speed  and  Velocity 57 

39.  Table  of  Volume  and  Capacity 5" 

40.  Table  of  Weight  or  Mass 58 

41 .  Table  of  Energy 58 

42.  Table  of  Pressure 59 

43.  Table  of  Power 59 

44.  Explanations 60 

45.  Exercises 61 

Chapter  7 
MISCELLANEOUS   TABLES 

Art.  46.  Table  of  M.^thematical  Constants 64 

47.  Table  of  Decimal  Eqttivalents  of  Common  Fractions  64 

48.  Table  of  Hyperbolic  Function's 65 

49.  Table  of  Napierian  Logarithms 66 

50.  Table  of  Multipliers  for  Transferring  Logarithms  66 

51 .  Explanations 66 

52.  Exercises 67 


Chapter  1 
GENERAL  EXPLANATIONS 


8  General  Explanations 

1.  Arguments  and  Functions 
Arguments  and  Functions  are  the  two  kinds  of  numbers  that 
appear  in  a  table,  the  former  being  the  numbers  which  are  given 
and  the  latter  those  which  are  sought.  The  arguments  are  the 
numbers  for  which  the  values  of  the  functions  have  been  com- 
puted; thus  in  v^,  values  of  7i  are  the  arguments  and  those  of 
Vn  are  the  functions.  An  argument  is  at  the  side  of  the  table,  or 
sometimes  part  of  it  is  at  the  side  and  part  at  the  top  or  foot; 
thus  when  the  square  of  6.48  is  sought  from  Table  7,  the  num- 
ber 6.4  is  found  at  the  left-hand  side  of  the  table  and  the  S  at  the 
top  or  foot,  then  at  the  intersection  of  the  horizontal  row  and  the 
vertical  column  is  found  the  function  or  number  41.99  which  is 
the  square  of  6.48  correct  to  four  places.  In  the  tables  of  this 
book  the  arguments  are  generally  in  bold-face  typo  and  the  func- 
tions in  common  type.  V 

2.  Precision  of  Tabular  Values 

The  values  of  the  functions  in  many  of  the  tables  of  this  book 
are  generally  given  only  to  four  significant  figures;  thus  the  square 
of  7.54  is  given  as  56.85,  although  its  exact  value  is  56.8516.  The 
greater  part  of  the  computations  in  physics  and  engineering 
require  only  three  or  four  significant  figures  to  be  determined 
with  precision,  since  the  given  data  rarely  extend  with  accuracy 
to  a  greater  number  of  figures. 

,  Significant  figures  in  a  number  are  those  not  preceded  by 
ciphers  after  a  decimal  point.  For  example,  each  of  the  numbers 
4507,  4.507,  0.04507  and  0.0004507  has  four  significant  figures; 
the  number  30.2734  has  six  significant  figures,  while  0.065  and  6.5 
have  only  two. 

In  any  table  the  last  figure  of  the  argument  is  exact,  but  the 
last  figure  of  the  function  is  liable  to  an  error.  Thus,  when  the 
computed  value  of  a  function  is  42.7854,  the  value  given  in  a  four- 
place  table  is  42.79,  the  last  figure  having  here  an  error  of  nearly 
one-half  of  a  unit;  when  the  computed  value  is  3.7851  the  value 
given  in  a  four-place  table  is  3.785,  the  last  figure  having  here  an 


General  Explanations  9 

error  of  one-tenth  of  a  unit.  The  maximum  error  in  a  tabulated 
function  is  hence  one-half  a  unit  in  the  last  figure,  and  the  prob- 
able error  is  one-fourth  of  a  unit. 

3.  Interpolation 

Interpolation  is  the  process  of  finding  the  value  of  a  function 
when  the  given  argument  lies  between  two  tabular  arguments. 
This  is  generall}'  done  by  regarding  the  function  as  varying  uni- 
formlj'  between  the  two  adjacent  tabular  values.  For  example,  if 
it  be  required  to  find  the  reciprocal  of  0.26-15  from  Table  G,  it  is 
seen  that  the  reciprocals  of  0.264  and  0.265  are  3.788  and  3.774; 
hence  the  reciprocal  of  0.2645  is  half-way  between  these,  or  3.781. 
Again,  let  it  be  required  to  find  the  square  root  of  85.04  from 
Table  8;  the  square  roots  of  85.0  and  85.1  are  found  to  be  9.220 
and  9.225;  the  difference  of  these  is  0.005,  and  0.4X0.005  is  0.002; 
then  the  square  root  of  85.04  is  9.220+0.002  =  9.222.  After  a  little 
practise  interpolation  in  a  four-place  table  ican  be  made  mentally. 

As  another  example,  let  it  be  required  to  find  the  value  of  sin 
43°  4'  from  Table  25;  here  the  sines  of  43°  0'  and  43°  10'  are 
0.68200  and  0.68412,  the  difference  of  wliich  is  0.00212,  thus  the 
difference  for  1'  is  0.000212  and  for  4'  it  is  0.00085;  then  sin  43°  4' 
is  0.68200+0.00085=0.68285.  When  the  function  decreases 
as  the  argmnent  increases,  the  computed  difference  is  to  be  sub- 
tracted from  the  greater  value  of  the  function;  thus  to  find  the 
reciprocal  of  0.5427,  the  reciprocals  of  0.542  and  0.513  are 
1.845  and  1.842;  the  difference  of  these  is  0.003,  and  0.003X0.7 
=  0.002;  hence  the  reciprocal  of  0.5427  is  1.845  -  0.002  =  1.843. 

The  precision  of  an  interpolated  value  is  less  than  that  of 
the  tabular  values,  because  the  assumption  that  the  function 
varies  uniformily  between  the  two  adjacent  tabular  values  is  not 
strictly  correct,  and  because  it  is  obtained  by  taking  the  differ- 
ence of  two  tabular  values  which  are  not  exact  in  the  last  figure. 
In  general  the  probable  error  in  an  interpolated  value  is  at  least 
one-haff  a  unit  in  the  last  figure. 

Inverse  interpolation  is  the  process  of  obtaining  an  argument 


10  General  Explanations 

from  a  given  function  when  the  latter  lies  between  two  adjacent 
tabular  values;  lliis  will  be  explained  in  Arts.  28  and  34, 

4.  Precision  of  Computed  Results 
It  is  important  that  a  computer  should  use  the  tables  so  as  to 
obtain  the  most  precise  result  possible  and  also  that  he  should 
not  attribute  to  the  result  a  precision  which  does  not  exist.  In 
general  no  more  tlian  four  significant  figures  can  ])C'  obtained  from 
a  four-place  table,  and  in  the  case  of  extended  computations  the 
last  figure  may  be  liable  to  an  error  of  one  unit.  For  example, 
the  value  of  VQ.S"-+8.r-  is  found  by  the  help  of  Tables  7  and  8 
to  be  10.50,  which  is  exact,  but  the  value  of  (1.25-+ 1. 45-) ^  is 
found  to  be  13.44,  which  is  one  unit  in  error  in  the  last  figure. 

Inexperienced  computers  sometimes,  in  making  interpola- 
tions, use  all  the  figures  obtained  in  the  multiplication  of  differ- 
ences, and  thus  carry  the  work  several  places  further  than  the 
tabular  values  warrant.  This  procedure  not  only  entails  additional 
work  and  gives  extra  figures  which  are  wholly  inaccurate,  but  it 
leads  the  computer  to  suppose  tliat  his  results  have  a  far  higher 
degree  of  precision  than  is  actually  the  case,  hence  vitiating  his 
judgment  and  perhaps  leading  to  the  deceit  of  others  as  well  as 
of  himself.  In  no  case  should  more  significant  figures  appear  in 
the  final  results  than  are  given  in  the  tables  which  are  used. 

5.  Exercises  for  Students 

1.  From  Table  G  olituin  the  reciprocals  of  0.7G,  0.7G5,  0.766, 
0.7665,  0.7668,  also  the  reciprocals  of  5.5,  5.53,  5.534,  5.536. 

2.  From  Table  7  ohtain  the  t'our-iilace  squares  of  1.85  and  1.86; 
verify  the  results  by  actual  iimltiplications. 

3.  From  Table  7  obtain  the  squares  of  8.5,  8.53,  8.534;  also  of 
0.85,  0.853,  0.8534. 

4.  Find  the  sine  of  18°  23'  from  Table  25. 

5.  Find  the  value  of  (3.27=  +  4.]82)2  by  the  help  of  Table  7;  com- 
pare it  with  the  resnlt  obtained  by  actual  multiplication. 

0.  Find  the  square  roots  of  6.35  and  63.5  from  Table  8;  also  the 
square  roots  of  6.352  and  65.32. 

7.  Find  the  values  of  1.42^  and  1.524^  from  Table  11. 


Chapter  2 
ARITHMETICAL  TABLES 


12 

Explanation  on  p.  38 


Arithmetical  Tables 


6.  Reciprocals 


n 

012345678             9 

O.IO 

10.00 

9.901 

9.804 

9-709 

9-615 

9.524 

9-434 

9-344, 9-259 

9-174 

O.II 

9.091 

9.009 

8.929 

8.850 

8.772 

8.696 

8.621 

8.547 

8.475 

8.403 

0.12 

8-333 

8.264 

8.197 

8.130 

8.065 

8.000 

7-937 

7-874 

7-813 

7752 

O.I3 

7.692 

7-634 

7-576 

7-519 

7  -  463 

7  407 

7-353 

7-299 

7.246 

7- 194 

0.14    7-143 

1 

7.092 

7.042 

6-993 

6.944 

6.897 

6.849 

6.803 

6.757 

6. 711 

o.is    6.667 

6.623 

6.579 

6.536 

6.494 

6.452 

6-410 

6.369 

6.329 

6.289 

0. 16 

6.250 

6. 211 

6.173 

6.135 

6.098 

6.061 

6.024 

5.988 

S-952 

5-917 

0.17 

5-882 

S-848 

5-814 

5-780 

5-747 

5-714 

5.682 

5-650 

5.618 

5-587 

0.18 

5-556 

5-525 

5 -495 

5-464 

5-435 

5-405 

5-376 

5-348 

5-319 

5-291 

0.19 

S-263 

5-236 

5.208 

5-181 

5-155 

5.128 

5- 102 

5-076 

5-051 

5-025 

0.20 

5.000 

4-975 

4-950 

4.926 

4.902 

4.878 

4-854 

4-831 

4.808 

4-785 

0.21 

4.762 

4-739 

4-717 

4.695 

4-673 

4.651 

4-630 

4.608 

4.587 

4.566 

0.22  ]4.545 

4-525 

4-505 

4.484 

4.464 

4-444 

4-425 

4.405 

4.386 

4.367 

0.23 

4   348 

4-329 

4.310 

4.292 

4-274 

4-255 

4-237 

4.219 

4.  202 

4.184 

0.24 

4.167 

4.149 

4-132 

4-iiS 

4.098 

4.082 

4.065 

4.049 

4032 

4.016 

0.25 

4.000 

3-984 

3-968 

3-953 

3-937 

3-922 

3-906 

3-891 

3-876 

3.861 

0.26 

3.846 

3-831 

3-817 

3.802 

3.788 

3-774 

3-759 

3-745 

3-731 

3-717 

0.27 

3-704 

3.690 

3-676 

3-663 

3-650 

3  636 

3-623 

3.610 

3-597 

3-584 

0.28 

3-571 

3-559 

3-546 

3-534 

3-521 

3  ■  509 

3-497 

3-484 

3472 

3.460 

0.29 

3-448 

3-436 

3-425 

3-413 

3-401 

3-390 

3-378 

3-367 

3  356 

3-344 

0.30 

3-333 

3-322 

3-3^^ 

3-300 

3.289 

3-279 

3.268 

3-257 

3-247 

3-236 

0.31 

3-226 

3-215 

3-205 

3-195 

3-185 

3-175 

3- "65 

3-155 

3-145 

3-135 

0.32 

3-125 

3-II5 

3.106 

3.096 

3.086 

3-077 

3-067 

3-058 

3-049 

3.040 

0.33 

3-030 

3.021 

3.012 

3-003 

2.994 

2-985 

2.976 

2.967 

2-959 

2-950 

0.34 

2.941 

2.933 

2.924 

2.915 

2.907 

2.899 

2.890 

2.882 

2.874 

2.865 

0.35 

2.857 

2.849 

2.841 

2.833 

2.825 

2.817 

2.809 

2.801 

2.793 

2.786 

0.36 

2.778 

2.770 

2.7.62 

2.755 

2.747 

2.740 

2.732 

2.725 

2.717 

2.710 

0.37 

2.703 

2.695 

2:688 

2 .  68-1 

2.674 

2.667 

2.660 

2.653 

2.646 

2.639 

0.38 

2.632 

2.625 

2.618 

2. 611 

2.604 

2-597 

2.591 

2.584 

2.577 

2.571 

0.39 

2.564 

2.558 

2.551 

2-545 

2.538 

2.532 

2.525 

2.519 

2.513 

2.  506 

0.40 

2.  500 

2.494 

2.488 

2.481 

2-475 

2.460 

2 .  463 

2.457 

2.45' 

2.445 

0.41 

2.439 

2.433 

2.427 

2.421 

2.415 

2.410 

2.404 

2-398 

2.392 

2.387 

0.42 

2.381 

2-375 

2.370 

2  -  364 

2.358 

2.353 

2.347 

2.342 

2 .  336 

2.331 

0.43 

2.326 

2.320 

2.315 

2.309 

2.304 

2.  299 

2.294 

2.288 

2 .  2S3 

2.278 

0.44 

2.273 

2.268 

2.  262 

2-257 

2.252 

2.247 

2.242 

2.237 

2.232 

2.227 

0.45 

2.  222 

2.217 

2.  212 

2.  208 

2.203 

2. 198 

2.193 

2.188 

2.183 

2.179 

0.46 

2.  174 

2. 169 

2. 165 

2. 160 

2.15s 

2. 151 

2. 146 

2. 141 

2.137 

2.132 

0.47 

2.128 

2.123 

2. 119 

2.  1 14 

2.  no 

2. 105 

2.  lOI 

2.096 

2.092 

2.p88 

0.48 

2.083 

2.079 

2.075 

2.070 

2.066 

2.062 

2.058 

2.053 

2  049 

2.045 

0.49 

2.041 

2-037 

2-033 

2.028 

2.024 

2.020 

2.0l6 

2.012 

2.008 

2.004 

0.50 

2.000 

1.996 

r.992 

1.9S8 

1.984 

1.980 

1.976 

1.972 

1.969 

1.965 

0.51 

1. 96 1 

1-957 

1-953 

1.949 

1.946 

1.942 

I  •  938 

I  •  934 

1.93' 

1.927 

0.52 

1-923 

1. 919 

1. 916 

I. 912 

1.908 

1 .  905 

1. 901 

1 .  89S 

1.894 

1.890 

0.53 

1.887 

1 .  883 

1.880 

1.876 

1-873 

1.869 

1.866 

1.862 

1 .  859 

1-855 

0.54 

1.852 

1.848 

1.84s 

1.842 

1.838 

1-835 

1.832 

1.828 

1.825 

1. 821 

n 

0123456789 

of  Numbers 


Arithmetical  Tables 


13 


n 

012345678             9 

0.55 

I. 818 

I. 815 

I. 812 

1.808 

1.805 

1.802 

1.799 

1.795 

1.792 

1.789 

0.56 

1.786 

1.783 

1-779 

1.776 

1-773 

1.770 

1.767 

1.764 

1.761 

I. 757 

0.57 

r-754 

1-75- 

1.748 

1-745 

1.742 

1-739 

1-736 

1-733 

1-730 

1.727 

0.58 

1.724 

I.  721 

I. 718 

I. 715 

I. 712 

1.709 

1.  706 

1.704 

1 .  701 

1.698 

0.59 

1-695 

1.692 

1.689 

1.686 

1.684 

1. 681 

1.678 

1-675 

1.672 

1.669 

0.60 

r.667 

1 .664 

1. 661 

1.658 

1.656 

1-653 

1.650 

1.647 

I  ■  645 

1.642 

0.61 

1-639 

1.637 

1-634 

I. 631 

1.629 

1.626 

1.623 

1.621 

1.618 

1. 616 

0.62 

1.613 

1. 610 

1.608 

1 .  605 

1.603 

1 .  600 

1-597 

1-595 

1.592 

1.590 

0.63 

1.587 

1-585 

1.582 

1.580 

1.577 

I-57S 

1-572 

1-570 

1.567 

1.565 

0.64 

1.562 

1.560 

1-558 

1. 555 

1.553 

r-550 

1-548 

1-546 

1-543 

1. 541 

0.65 

1.538 

1.536 

1-534 

1. 531 

1.529 

1-527 

1-524 

1.522 

1.520 

1-517 

0.66 

1.515 

1.513 

I. 511 

1.508 

1.506 

1.504 

1.502 

1.499 

1-497 

1.495 

0.67 

1-493 

1.490 

1.488 

1.486 

1.484 

1. 481 

1-479 

1.477 

1-475 

1.473 

0.68 

I. 471 

1.468 

1 .466 

1.464 

1.462 

1.460 

1-458 

1-456 

1-453 

1. 451 

0.69 

1.449 

1-447 

1.445 

1.443 

1-441 

1-439 

1-437 

1-435 

1.433 

I-431 

0.70 

1.429 

1.427 

1.425 

1.422 

1.420 

I. 418 

1 .416 

1.414 

1.412 

1.410 

0.71 

1.408 

1.406 

1.404 

1.403 

1. 401 

1-399 

1-397 

1-395 

1.393 

1-391 

0.72 

1.389 

1-387 

1.385 

1.383 

1-381 

1-379 

1.377 

1-376 

1.374 

1-372 

0.73 

1.370 

1.368 

1.366 

1.364 

1.362 

1. 361 

1-359 

1-357 

1-355 

1-353 

0.74 

1-351 

1.350 

1.348 

1.346 

1.344 

1-342 

1-340 

1-339 

1-337 

^■ZiZ 

0.75 

1-333 

-i-iS^ 

1.330 

1.32S 

1.326 

1-325 

1-323 

1.321 

1-319 

1-318 

0.76 

1.316 

r.314 

1. 312 

1. 311 

1.309 

1-307 

I  -  305 

1-304 

1.302 

1.300 

0.77 

1.299 

1.297 

r.295 

1.294 

1.292 

1 .  290 

1.289 

1.287 

1.285 

1.284 

0.78 

1.282 

1.280 

1.279 

1.277 

1.276 

1.274 

I.  272 

I. 271 

1.269 

1.267 

0.79 

1.266 

1.264 

1.263 

I. 261 

1.259 

1.258 

1.256 

1-255 

1-253 

1 .  252 

0.80 

1.250 

1.248 

1.247 

1.245 

1.244 

r.  242 

1.241 

1.239 

1-238 

1.236 

0.81 

1-235 

1.233 

1.232 

1.230 

1 .  229 

1.227 

1.225 

1.224 

1 .  222 

1 .  221 

0.82 

1.220 

I. 218 

I. 217 

1.215 

I.  214 

I. 212 

1.211 

1.  209 

1.208 

1 .  206 

0.83 

1.205 

1.203 

1 .  202 

1 .  200 

1. 199 

I.  198 

1. 196 

I -195 

1-193 

1. 192 

0.84 

r.  190 

1. 189 

1. 188 

1. 186 

1. 185 

I.  183 

1. 182 

1.181 

1.179 

1.178 

0.85 

1. 176 

I.I75 

1. 174 

1 .  172 

1. 171 

1. 170 

1.168 

1. 167 

1.166 

1. 164 

0.86 

1. 163 

1. 161 

1 .  160 

1. 159 

1-157 

1. 156 

I-IS5 

I -153 

1.152 

1.151 

0.87 

1. 149 

1. 148 

1. 147 

1. 145 

I -144 

1. 143 

1. 142 

1. 140 

1-139 

1. 138 

0.88 

1. 136 

I.I35 

1. 134 

1. 133 

1. 131 

I.  130 

1.129 

1. 127 

1. 126 

1.125 

0.89 

1. 124 

1 .  122 

1 .  121 

1 .  120 

r.  119 

1. 117 

1. 116 

1.115 

1. 114 

1. 112 

0.90 

I.  Ill 

I .  no 

1. 109 

1. 107 

1 .  106 

r.  105 

1. 104 

I.  103 

I.  lOI 

1. 100 

0.91 

1.099 

1.098 

1 .  096 

1.095 

1.094 

1.093 

1.092 

1. 091 

1.089 

1.088 

0.92 

1.087 

1.086 

1.085 

1 .  083 

1.082 

1. 081 

1.080 

1.079 

1.078 

1.076 

0-93 

1-075 

1.074 

1.073 

1 .072 

r.071 

1.070 

1.068 

1.067 

1.066 

1.065 

0.94 

1 .064 

1.063 

1 .062 

1 .060 

1-059 

1.058 

1.057 

1.056 

1-055 

1.054 

0.95 

1.053 

1.052 

1.050 

1.049 

1 .048 

1.047 

1.046 

1-045 

1.044 

1 .043 

0.96 

1.042 

1 .041 

1 .040 

1.038 

1.037 

1.036 

1.035 

1.034 

1.033 

1.032 

0.97 

1. 031 

1.030 

1.029 

1.028 

1 .027 

1.026 

1.025 

1.024 

1 .022 

1 .  021 

0.98 

1 .020 

I  .oig 

1. 018 

1. 017 

1. 016 

1. 015 

1 .014 

1. 013 

1 .012 

I .  on 

0.99 

I. 010 

1.009 

1.008 

1.007 

1 .006 

1 .005 

1 .004 

1.003 

1.002 

1. 001 

» 

012345678             9 

14 


Arithmetical  Tables 


7.  Squares  of  Num- 


n 

I.O 

012345678             9 

1. 000 

1 .020 

1 .040 

1. 061 

1.082 

1.103 

1. 124 

I -145 

1.166 

1.188 

i.i 

1 .  210 

1.232 

1.254 

1.277 

1.300 

1-323 

1.346 

1-369 

1-392 

1. 416 

1.2 

1.440 

1.464 

1.488 

1-513 

1-538 

1-563 

1.588 

1.613 

1.638 

1.664 

1-3 

1 .  690 

I.  716 

1.742 

1.769 

1.796 

1.823 

1.850 

1.877 

1.904 

1-932 

1.4 

1.960 

1.988 

2.016 

2.045 

2.074 

2.103 

2.132 

2.161 

2. 190 

2.220 

1-5 

2.  250 

2.280 

2.310 

2.341 

2.372 

2.403 

2.434 

2.465 

2.496 

2.528 

1.6 

2.  560 

2.592 

2.  624 

2.657 

2.690 

2.723 

2.756 

2.789 

2.822 

2.856 

1.7 

2.890 

2.924 

2-958 

2.993 

3.028 

3-063 

3.098 

3--i3,3 

3-168 

3-204 

1.8 

3-240 

3-276 

3-312 

3-349 

3-386 

3-423 

3.460 

3-497 

3-534 

3-572 

1-9 

3.610 

3-648 

3.686 

3-725 

3-764 

3  803 

3-842 

3-881 

3.920 

3.960 

2.0 

4.000 

4.040 

4.080 

4. 121 

4. 162 

4.203 

4.244 

4.285 

4.326 

4.368 

2.1 

4.410 

4.452 

4.494 

4-537 

4.580 

4.623 

4.666 

4-709 

4.752 

4.796 

2.2 

4. 840 

4.884 

4.928 

4.973 

5-018 

5-063 

5.108 

5- 153 

'5-198 

5-244 

2.3 

5.290 

5-336 

5-382 

5-429 

5-476 

5-523 

5-570 

5-617 

5-664 

5-712 

2.4 

5-760 

5.808 

5 -856 

5-905 

5-954 

6.003 

6.052 

6.  loi 

6.150 

6.200 

2.5 

6.250 

6.300 

6-350 

6.401 

6.452 

6.503 

6-554 

6.605 

6.656 

6.708 

2.6 

6.  760 

6.812 

6.864 

6.917 

6.970 

7-023 

7.076 

7.129 

7.182 

7-236 

2.7 

7.290 

7-344 

7-398 

7-453 

7-508 

7-563 

7.618 

7-673 

7.728 

7-784 

2.8 

7.840 

7.896 

7-952 

8.009 

8.066 

8.123 

8.180 

8.237 

8.294 

8-352 

2.9 

8.410 

8.468 

8.526 

8.585 

8.644 

8.703 

8.762 

8.821 

8.880 

8.940 

3-0 

9.000 

9.060 

9. 120 

9.181 

9.242 

9-303 

9-364 

9-425 

9.486 

9-548 

3.1 

9.610 

9-672 

9-734 

9-797 

9.860 

9-923 

9.986 

10.05 

10. 11 

10.18 

3.2 

10.24 

10.30 

10.37 

10.43 

10.50 

10.56 

10.63 

10.69 

10.  76 

10.82 

33 

10.89 

10.96 

11.02 

11.09 

11.16 

11.22 

11.29 

11.36 

11.42 

11.49 

3-4 

11.56 

11.63 

II.  70 

11.76 

11.83 

11.90 

11.97 

12.04 

12. 11 

12.18 

35 

12.25 

12.32 

12.39 

12.46 

12.53 

12.60 

12.67 

12.74 

12.82 

12.89 

3-6 

12.96 

13-03 

13.10 

13-18 

13-25 

13-32 

13-40 

13-47 

13-54 

13.62 

3-7 

13-69 

13-76 

13-84 

13-91 

13-99 

14.06 

14-14 

14.  21 

14.29 

14-36 

3-8 

14.44 

14-52 

14-59 

14.67 

14-75 

14. 82 

14.90 

14.98 

15-05 

15-13 

3-9 

15.21 

15-29 

15-37 

15-44 

15-52 

15.60 

15.68 

15-76 

15-84 

15-92 

4.0 

16.00 

16. oS 

16.16 

16.  24 

16.32 

16.40 

16.48 

16.56 

16.65 

16.73 

4.1 

16.81 

16.89 

16.97 

17.06 

17.14 

17.  22 

17-31 

17-39 

17-47 

17-56 

4.2 

17.64 

17.72 

17.81 

17.89 

17.98 

18.06 

18.15 

18.23 

18.32 

18.40 

4-3 

18.49 

18.58 

18.66 

18.75 

18.84 

18.92 

19.01 

19. 10 

19.  18 

19.27 

4-4 

19-36 

19-45 

19-54 

19.62 

19.71 

19.80 

19.89 

19.98 

20.07 

20. 16 

4-5 

20.  25 

20.34 

20.43 

20.52 

«o.6i 

20.  70 

20.79 

20.88 

20.981 

21.07 

4.6 

21. 16 

21.25 

21.34 

21.44 

21-53 

21.62 

21.72 

21. Si 

21.90 

22.00 

4-7 

22.09 

22.18 

22.28 

22.37 

22.47 

22.56 

22.66 

22.75 

22.8_S 

22.94 

4.8 

23-04 

23-14 

23   23 

23-33 

23-43 

23-52 

23. 62 

23-72 

23.81 

23-91 

4-9 

24.01 

24.  II 

24.21 

24-30 

24.40 

24.50 

24.60 

24.70 

24.80 

24-90 

50 

25.00 

25.10 

25.20 

25-30 

25-40 

25-50 

25.60 

25.70 

25.81 

25-91 

5-1 

26.01 

26.  II 

26.21 

26.32 

26.42 

26.  52 

26.63 

26.73 

26.83 

26.94 

5-2 

27.04 

27.14 

27-25 

27-35 

27.46 

27-56 

27.67 

27-77 

27.88 

27.98 

53 

28.09 

28.20 

28.30 

28.41 

28.52 

28.62   28.73 

28.84 

28. 94 

29.05 

5-4 

.?9.  16 

29-27 

29 -38 

29.48 

29-59 

29.70   29.81 

29.92 I30.03 

30. 14 

n 

0             12345678             9 

Arithmetical  Tables 
bers  from  1.00  to  9.99 


15 


n 

c 

>     I 

2 

3     4 

5' 

5     7     8 

9 

S-S 

30 

25 

30 

36 

30 

47 

30.58 

30.69 

30 

80 

30 

91 

31.02 

31-  14 

31 

25 

5.6 

31 

36 

31 

47 

31 

58 

31.70 

31.81 

31 

92 

32 

04 

32.15 

32.26 

32 

38 

5-7 

32 

49 

32 

60 

32 

72 

32.83 

32-95 

33 

06 

33 

18 

33-29 

33.41 

33 

52 

5.8 

33 

64 

33 

76 

33 

87 

33-99 

34-11 

34 

22 

34 

34 

34.46 

34.57 

34 

69 

5.9 

34 

81 

34 

93 

35 

05 

35.16 

35.28 

35 

40 

35 

52 

35  64 

35.76 

35 

88 

6.0 

36 

00 

36 

12 

36 

24 

36.36 

36.48 

36 

60 

36 

72 

36.84 

36-97 

31 

09 

6.1 

37 

21 

37 

33 

37 

45 

37.58 

37.7° 

37 

82 

37 

95 

38. 07 

38.19 

38 

32 

6.2 

38 

44 

38.56 

38.69 

38.81 

38-94 

39 

06 

39 

19 

39-31 

39-44 

39 

56 

6.3 

39 

69 

39 

82 

39 

94 

40.07 

40.  20 

40 

32 

40 

45 

40.58 

40.70 

40 

83 

6.4 

40 

96 

41 

09 

41 

22 

41.34 

41.47 

41 

60 

41 

73 

41.86 

41.99 

42 

12 

6.5 

42 

25 

42 

38 

42 

51 

42.64 

42-77 

42 

90 

43 

03 

43.16 

43.30 

43 

43 

6.6 

43 

56 

43 

69 

43 

82 

43.96 

44.09 

44 

22 

44 

36 

44.49 

44.62 

44 

76 

6.7 

44 

89 

45 

02 

45 

16 

45-29 

45-43 

45 

56 

45 

70 

45.83 

45-97 

46 

10 

6.8 

46 

24 

46.38 

46 

51 

46.65 

46.79 

46 

92 

47 

06 

47.20 

47.33 

47 

47 

6.9 

47 

61 

47 

75 

47 

89 

48.02 

48.16 

48 

30 

48 

44 

48.58 

48.72 

48 

86 

7.0 

49 

00 

49 

14 

49 

28 

4942 

49  •?6 

49 

70 

49 

84 

49.98 

50.13 

50 

27 

7.1 

50 

41 

SO 

55 

50 

69 

50.84 

50.98 

SI 

12 

51 

27 

51.41 

51.55 

51 

70 

7.2 

SI 

84 

51 

98 

52 

13 

52.27 

52-42 

52 

56 

52 

71 

52.85 

53-00 

53 

14 

7.3 

53 

29 

53 

44 

53 

58 

53.73 

.53-88 

54 

02 

54 

17 

S4-r32 

54-46 

54 

61 

7.4 

54 

76 

54 

91 

55 

06 

5S-20 

55-35 

55 

50 

55 

65 

55-80 

55-95 

56 

10 

7.5 

56 

25 

56 

40 

56 

55 

56.70 

56.85 

57 

00 

57 

15 

57-30 

57.46 

57 

61 

7.6 

57 

76 

57 

91 

58 

06 

58-22 

58.37 

58 

52 

58 

68 

58.83 

58.98 

59 

14 

7.7 

59 

29 

59 

44 

59 

60 

59.75 

59.91 

60 

06 

60 

22 

60.37 

60.53 

60 

68 

7.8 

60 

84 

61 

00 

61 

15 

61.31 

61.47 

61 

62 

61 

78 

61.94 

62.09 

62 

25 

7.9 

62 

41 

62 

57 

62 

73 

62.88 

63.04 

63 

20 

63 

36 

63.52 

63.68 

63 

84 

8.0 

64 

00 

64 

16 

64 

32 

64.48 

64. 64 

64 

80 

64. 96 

65.12 

65-29 

65 

45 

8.1 

65 

61 

65 

77 

65 

93 

66. 10 

66.26 

66 

42 

66 

59 

66.75 

66.91 

67 

08 

8.2 

67 

24 

67 

40 

67 

57 

67.73 

67.90 

68 

06 

68 

23 

68.39 

68.56 

68 

72 

8.3 

68 

89 

69 

06 

69 

22 

6939 

69-56 

69 

72 

69.89 

70.06 

70.  22 

70 

39 

8.4 

70 

56 

70 

73 

70 

90 

71.06 

71.23 

71 

40 

71 

57 

71.74 

71.91 

72 

08 

8.5 

72 

25 

72 

42 

72 

59 

72.76 

72.93 

73 

10 

73 

27 

73.44 

73.62 

73 

79 

8.6 

73 

96 

74 

13 

74 

30 

74.48 

74.65 

74 

82 

75 

00 

75.17 

75. '34 

75- 

52 

8.7 

75 

69 

75 

86 

76 

04 

76.21 

76.39 

76 

56 

76 

74 

76-91 

77.09 

77 

26 

8.8 

77 

44 

77 

62 

77 

79 

77.97 

78.15 

78 

32 

78 

SO 

78.68 

78.85 

79 

03 

8.9 

79 

21 

79 

39 

79 

57 

79.74 

79-92 

80 

10 

80 

28 

80.46 

80.64 

80. 

82 

9.0 

81 

00 

81 

18 

81 

36 

81.54 

81.72 

81 

90 

82 

08 

82.26 

82.45 

82. 

63 

9.1 

82 

81 

82 

99 

83 

1.7 

83.36 

83.54 

83 

72 

83 

91 

84.09 

84.27 

84.46] 

9-2 

84.64 

84 

82 

85 

01 

85.19 

85.38 

85 

56 

85 

75 

85.93 

86.12 

86. 

3° 

9.3 

86 

49 

86 

68 

86 

86 

87.05 

87.24 

87 

42 

87 

61 

87.80 

87.98 

88. 

17 

9.4 

88 

36 

88 

55 

88 

74 

88.92 

89.11 

89 

30 

89 

49 

89.68 

89.87 

90. 

06 

9.5 

90 

25 

90 

44 

90 

63 

90.82 

91.01 

91 

20 

91 

39 

91.58 

91.78 

91. 

97 

9.6 

92 

16 

92 

35 

92 

S4 

92.74 

92.93 

93 

12 

93 

32 

93.51 

93.70 

93. 

90 

9.7 

94 

09 

94 

28 

94 

48 

94.67 

94.87 

95. 

06 

95. 

26 

95.45 

95.65 

95- 

84 

9-8 

96 

04 

96 

24 

96 

43 

96.63 

96.83 

97 

02 

97 

22 

97.42 

97.61 

97- 

81 

9.9 

98. 

01 

98 

21 

98. 

41 

98.60 

98.80 

99- 

00 

99 

20 

99.40 

99.60 

99. 

80 

n 

0 

I 

2 

345678     < 

J 

16 


Arithmetical  Tables 


8.   Square  Roots  of 


rt 

I.O 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

1. 000 

1.005 

1.010 

1.015 

1 .020 

1.025 

1.030 

1-034 

1-039 

1.044 

I.l 

1.049 

I -054' 

1.05S 

1.063 

1.068 

1.072 

1.077 

1.0S2 

1.086 

1. 091 

1.2 

I.095 

1. 100 

1. 105 

I.  109 

1 .  114 

1.118 

1. 122 

1. 127 

1.131 

1.136 

1-3 

1. 140 

I -145 

1.149 

I.  153 

1.158 

1. 162 

1.166 

1. 170 

1.175 

1.179 

1.4 

1. 183 

1.187 

1. 192 

1.196 

1 .  200 

1 .  204 

1.208 

1.212 

1.217 

1.221 

1.5 

1.225 

1.229 

1.233 

1.237 

1.241 

1.245 

1.249 

1.253 

1.257 

1. 261 

1.6 

1.265 

1.269 

1.273 

1.277 

1.281 

1.285 

1.288 

1.292 

1.296 

1.300 

1.7 

1.304 

1.308 

1.3" 

1.315 

1.319 

1.323 

1.327 

1.330 

1-334 

1.338 

1.8 

1.342 

1.345 

1.349 

1-353 

1.356 

1.360 

1.364 

1.367 

1-371 

1.37s 

1.9 

1.378 

1.382 

1-386 

1.^89 

1.393 

1-396 

1.400 

1.404 

1.407 

1.411 

2.0 

1.414 

1. 418 

1. 421 

1-425 

1.428 

1-432 

1.435 

1.439 

1.442 

1.446 

2.1 

1.449 

1-453 

1.456 

1-459 

1.463 

1.466 

1.470 

1.473 

1.476 

1.480 

2.2 

1.483 

1-487 

1.490 

1-493 

1-497 

1.500 

1.503 

1.507 

1 .  510 

1.513 

2.3 

1-517 

1.520 

1.523 

1.526 

1-530 

1-533 

1.536 

1.539 

1-543 

1-546 

2.4 

1-549 

1-552 

1-556 

1-559 

1.562 

1.565 

1.568 

1.572 

1-575 

1-578 

2.5 

I. 581 

1-584 

1-587 

I -591 

1-594 

1.597 

1.600 

1.603 

1.606 

1.609 

2.6 

I. 612 

1. 616 

1 .619 

1.622 

1.625 

1.628 

1. 631 

1.634 

1-637 

1 .640 

2.7 

1.643 

1.646 

1.649 

1.652 

1-655 

1.658 

i.66i 

1.664 

1.667 

1.670 

2.8 

1-673 

1.676 

1.679 

1.682 

1.685 

1.688 

1.691 

1.694 

1-697 

1.700 

2.9 

i-7°3 

1.706 

1.709 

1 .  712 

1-715 

1.71S 

I.  720 

1.723 

1.726 

1.729 

3.0 

1-732 

1-735 

1.738 

1.741 

1-744 

1.746 

1.749 

1.752 

I-75S 

1-758 

3.1 

I.  761 

1-764 

1.766 

1.769 

1.772 

1-775 

1.778 

1.780 

1.783 

1.786 

3-2 

1.789 

1.792 

1-794 

1.797 

1.  800 

1.803 

I.  806 

1.808 

1.811 

1.814 

3-3 

I. 817 

I. 819 

1.822 

1.825 

1.828 

1.830 

1-833 

1.836 

1.838 

1.841 

3.4 

1.844 

1.847 

1.849 

1.852 

1.855 

1.857 

1.860 

1.863 

1.865 

1.868 

3-5 

1. 871 

1.873 

1.876 

1.879 

1.881 

1.884 

1.887 

1.889 

1.892 

1.89s 

3.6 

1.897 

1.900 

1-903 

1.905 

1.90S 

1. 910 

1. 913 

1. 916 

1.918 

1.921 

3.7 

1.924 

1.926 

1.929 

1. 931 

1-934 

1.936 

1.939 

1.942 

1.944 

1.947 

3.8 

1.949 

1-952 

1-954 

1-957 

1.960 

1.962 

1.965 

1.967 

1.970 

1.972 

3.9 

1-975 

1-977 

1.980 

1.982 

1-985 

1.987 

1.990 

1.992 

1.995 

1.997 

4.0 

2.000 

2.002 

2.005 

2.007 

2.010 

2.012 

2.015 

2.017 

2.020 

2.022 

4.1 

2.025 

2.027 

2.030 

2.032 

2-035 

2.037 

2.040 

2.042 

2.045 

2.047 

4.2 

2.049 

2.052 

2.054 

2.057 

2.059 

2.062 

2.064 

2.066 

2.069 

2.071 

4-3 

2.074 

2.076 

2.078 

2.081 

2.083 

2.086 

2.088 

2.090 

2.093 

2.09s 

4.4 

2.098 

2. 100 

2. 102 

2.105 

2. 107 

2. 110 

2.  112 

2. 114 

2.117 

2. 119 

4.5 

2. 121 

2. 124 

2. 126 

2.128 

2. 131 

2.133 

2-135 

2.138 

2. 140 

2. 142 

4.6 

2. 145 

2.147 

2.149 

2.152 

2.154 

2. 156 

2.159 

2. 161 

2.163 

2. 166 

4.7 

2.168 

2. 170 

2-173 

2.175 

2.177 

2.179 

2.182 

2.184 

2.186 

2.189 

4.8 

2. 191 

2-193 

2.195 

2.198 

2.  200 

2.202 

2.  205 

2.207 

2.  209 

-2.211 

4.9 

2.214 

2.216 

2.218 

2.220 

2.223 

2.225 

2.  227 

2.229 

2.232 

2.234 

5.0 

2.236 

2.238 

2.241 

2 .  243 

2.245 

2.247 

2.249 

2.252 

2.254 

2.256 

5.1 

2.258 

2.261 

2.263 

2.265 

2.267 

2.  269 

2.272 

2.274 

2.276 

2.278 

5-2 

2.280 

2.283 

2.2S5 

2.287 

2.2S9 

2.  291 

2.293 

2.296 

2.298 

2.300 

53 

2.302 

2.3°4 

2.307 

2.309 

2. 311 

2.313 

2.315 

2.317 

2.319 

2.322 

5-4 

11 

2.324 

2.326 

2.328 

2.330 

2.332 

2.33s 

2.337 

2.339 

2.341 

2.343 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Arithmetical  Tables 
Numbers  from  1.00  to  99.9 


17 

Continued  on  p.  18 


n 

01             23456789 

5.5 

2.345 

2-347 

2-349 

2-352 

2-354 

2-356 

2.358 

2.360 

2.362 

2-364 

5.6 

2.366 

2.369 

2.371 

2-373 

2-375 

2-377 

2.379 

2.381 

2-383 

2-385 

5.7 

2.387 

2.390 

2.392 

2-394 

2.396 

2.398 

2. 400 

2.402 

2.404 

2.406 

5.8 

2 .  408 

2.410 

2.412 

2.415 

2.417 

2.419 

2. 421 

2.423 

2.425 

2.427 

5-9 

2.429 

2.431 

2.433 

2.435 

2-437 

2.439 

2.441 

2.443 

2.445 

2.447 

6.0 

2.449 

2.452 

2-454 

2.456 

2.458 

2. 460 

2. 462 

2.464 

2.466 

2.468 

6.1 

2.470 

2.472 

2.474 

2.476 

2.478 

2 .  4S0 

2.482 

2.484 

2.486 

2.488 

6.2 

2.490 

2.492 

2.494 

2.496 

2.498 

2.  500 

2.  502 

2.504 

2.  506 

2 .  508 

6.3 

2.510 

2.512 

2.514 

2.516 

2.518 

2.520 

2.  522 

2.524 

2.526 

2.528 

6.4 

2-530 

2.532 

2.534 

2.536 

2.538 

2.540 

2.542 

2.544 

2.546 

2.548 

6.5 

2.550 

2.551 

2.553 

2.555 

2.557 

2.559 

2.561 

2-563 

2.565 

2.567 

6.6 

2.569 

2.571 

2.573 

2-575 

2.577 

2-579 

2.581 

2.583 

2.585 

2-587 

6.7 

2.588 

2.590 

2.592 

2.594 

2.596 

2-598 

2.600 

2.602 

2.604 

2.606 

6.8 

2.608 

2.610 

2.612 

2.613 

2.615 

2.617 

2.619 

2.621 

2.623 

2.625 

6.9 

2.627 

2.629 

2.631 

2.632 

2.634 

2.636 

2.638 

2.  640 

2.642 

2.644 

7.0 

2. 646 

2.648 

2.650 

2.651 

2.653 

2-655 

2.657 

2.659 

2.661 

2.663 

7-1 

2. 665 

2.666 

2.668 

2.  670 

2.672 

2.674 

2.676 

2.678 

2.680 

2.681 

7.2 

2.683 

2.685 

2.687 

2.689 

2.  691 

2.693 

2.694 

2.  696 

2.698 

2.  700 

7-3 

2.  702 

2.704 

2.  706 

2.707 

2.  709 

2.711 

2.713 

2-715 

2.717 

2.718 

7.4 

2.  720 

2.722 

2.724 

2.726 

2.728 

2.729 

2-731 

2-733 

2-735 

2.737 

7.5 

2.739 

2.740 

2.742 

2.744 

2.746 

2.748 

2.750 

2-751 

2.753 

2.755 

7.6 

2.757 

2.759 

2.  760 

2.762 

2.764 

2.766 

2.768 

2.  769 

2.771 

2-773 

7-7 

2-775 

2.777 

2.77S 

2.780 

2.782 

2.784 

2.786 

2.787 

2.789 

2.791 

7.8 

2-793 

2-795 

2.796 

2.79S 

2.  800 

2.802 

2.804 

2.805 

2.807 

2.809 

7.9 

2. 811 

2.S12 

2.814 

2.816 

2.818 

2.820 

2.821 

2.823 

2.825 

2.827 

8.0 

2.828 

2.830 

2.832 

2.834 

2.835 

2.837 

2.839 

2.841 

2.843 

2.844 

8.1 

2.846 

2.848 

2.850 

2.851 

2.853 

2.855 

2-857 

2.858 

2.860 

2.862 

8.2 

2.864 

2.865 

2.867 

2,869 

2.871 

2.872 

2.874 

2.876 

2.877 

2.879 

8.3 

2.881 

2.883 

2.884 

2.886 

2.888 

2.890 

2.891 

2.893 

2.895 

2.897 

8.4 

2.898 

2. 900 

2.  902 

2.903 

2.905 

2.907 

2.909 

2.910 

2 .  912 

2.914 

8.5 

2.915 

2.917 

2.919 

2.921 

2.  922 

2.924 

2.926 

2.927 

2.929 

2.931 

8.6 

2.933 

2-934 

2.936 

2.938 

2.939 

2.941' 

2.943 

2.944 

2.946 

2.948 

8.7 

2.950 

2.951 

2.953 

2.955 

2.956 

2.958 

2. 960 

2.961 

2.963 

2.965 

8.8 

2. 966 

2.96S 

2.970 

2.972 

2.973 

2-975 

2-977 

2.978 

2 .  9S0 

2.982 

8.9 

2.983 

2.985 

2.987 

2. 988 

2.  990 

2. 992 

2-993 

2-995 

2.997 

2.998 

9.0 

3.000 

3.002 

3.003 

3 .  005 

3-007 

3-008 

3.010 

3.012 

3.013 

3-oiS 

9.1 

3-017 

3.01S 

3.020 

3.022 

3-023 

3-025 

3-027 

3.028 

3-030 

3.032 

9.2 

3-033 

3-035 

3-03S 

3.038 

3.040 

3-041 

3 -043 

3  -  045 

3.046 

3.048 

9-3 

3-050 

3-051 

3-053 

3-055 

3-056 

3-058 

3.059 

3.061 

3-063 

3.064 

9-4 

3.066 

3.068 

3.069 

3-071 

3.072 

3-074 

3.076 

3-077 

3-079 

3-081 

9.5 

3.082 

3.084 

3-085 

3-087 

3.089 

3.090 

3-092 

3-094 

3-095 

3-097 

9.6 

3.098 

3.100 

3.102 

3-103 

3-105 

3.106 

3.108 

3.110 

3. Ill 

3-"3 

9-7 

3-114 

3. 116 

3. 118 

3-119 

3. 121 

3.122 

3-124 

3- 126 

3-127 

3.129 

9-8 

3- 130 

3-132 

3-134 

3-135 

3-137 

3-138 

3-140 

3-142 

3-143 

3- 145 

9-9 

3.146:3.148 

3-150 

3-151 

3-153 

3-154 

3-156 

3-158 

3-159 

3. 161 

n 

012             345678             9 

18 

Continued  from  p.  17 


Arithmetical  Tables 


Square  Roots  of 


n 

10 

.0          .1           .2           .3           -4          -5          -6          .7         -8           .9 

3.162 

3-178 

3-194 

3.209 

3-225 

3.240 

3.256 

3.271 

3.286 

3.302 

ZI 

3-317 

3-332 

3-347 

3.362 

3-376 

3.391 

3.406 

3.421 

3-435 

3.450 

12 

3-464 

3-479 

3-493 

3.507 

3-521 

3  536 

3.550 

3.564 

3578 

3-592 

13 

3.606 

3-619 

3-633 

3.647 

3-661 

3.674 

3-688 

3.701 

3-715 

3-728 

14 

3-742 

3-755 

3-768 

3.782 

3.795 

3.808 

3.821 

3.834 

3-847 

3-860 

IS 

3-873 

3-886 

3-899 

3.912 

3.924 

3.937 

3-950 

3.962 

3-975 

3-987 

i6 

4.000 

4.012 

4-025 

4.037 

4.050 

4.062 

4.074 

4.087 

4-099 

4.1H 

17 

4-123 

4.135 

4-147 

4.159 

4. 171 

4.183 

4-I9S 

4.207 

4.219 

4-231 

i8 

4-243 

4.254 

4.266 

4.278 

4.290 

4.301 

4.313 

4.324 

4-336 

4-347 

19 

4-359 

4.370 

4.382 

4.393 

4.405 

4.416 

4.427 

4.438 

4-450 

4.461 

20 

4-472 

4.483 

4-494 

4.506 

4.517 

4.528 

4.539 

4.550 

4-561 

4.572 

21 

4-583 

4.593 

4.604 

4-615 

4.626 

4.637 

4.648 

4.658 

4.669 

4.680 

22 

4.690 

4.701 

4.712 

4.722 

4.733 

4-743 

4.754 

4.764 

4.775 

4.785 

23 

4.796 

4.806 

4.817 

4.827 

4-837 

4.848 

4.858 

4.868 

4.879 

4.889 

24 

4.899 

4.909 

4.919 

4-930 

4.940 

4-950 

4.960 

4-970 

4.980 

4.990 

25 

5  .000 

5.010 

5  ■  020 

5-030 

5.040 

5.050 

5.060 

5-070 

5.079 

5.089 

26 

5-°99 

5.109 

5-II9 

5.128 

5-138 

5.148 

5-158 

5-167 

5.177 

5-187 

27 

5-196 

5.206 

5-215 

S-225 

5-235 

5-244 

5-254 

5-263 

5.273 

5.282 

28 

5-292 

5-301 

5-310 

5-320 

5-329 

5-339 

5-348 

5-357 

5.367 

5-376 

29 

5.385 

5-394 

5-404 

5.413 

5 -422 

5.431 

5 -441 

5-450 

5.459 

5.468 

30 

5-477 

5-486 

5-495 

5.505 

5-514 

5.523 

5-532 

5-541 

5.550 

5-559 

31 

5-568 

5-577 

5-586 

5-595 

5-604 

5.612 

5-621 

5-630 

5. 639 

5.648 

32 

5-657 

5.666 

5-675 

5.683 

5.692 

5.701 

5-710 

S-718 

5.727 

5-736 

33 

5-745 

5-753 

5.762 

5.771 

5-779 

5.788 

5-797 

5-805 

5.814 

5.822 

34 

5-831 

5.840 

5-848 

5-857 

5-865 

5.874 

5.882 

5. .89 1 

5.899 

5.908 

35 

5-916 

5-925 

5  -  933 

5-941 

5-950 

5.958 

5-967 

5-975 

5.983 

5-992 

36 

6.000 

6.008 

6.017 

6.025 

6 .  033 

6.042 

6.050 

6.058 

6.066 

6075 

37 

5.083 

6.091 

6.099 

6. 107 

6. 116 

6. 124 

6.132 

6. 140 

6.148 

6.156 

38 

6.164 

6-173 

6. 181 

6.189 

6.197 

6.20s 

6.213 

6.221 

6.229 

6-237 

39 

6.245 

6-253 

6.261 

6.269 

6.277 

6.28s 

6.293 

6.301 

6.309 

6-317 

40 

6-325 

6-332 

6.340 

6.348 

6.356 

6.364 

6.372 

6.380 

6.387 

6-395 

41 

6.403 

6. 411 

6.419 

6.427 

6.434 

6.442 

6.450 

6.458 

6.465 

6.473 

42 

6.481 

6.488 

6.496 

6.504 

6.512 

6.519 

6.527 

6-535 

6.542 

6.550, 

43 

6-557 

6-565 

6-573 

6.580 

6.588 

6.595 

6 .  603 

6. 611 

6.618 

6.626" 

44 

6-633 

6.641 

6.648 

6.656 

6.663 

6.671 

6.67S 

6.686 

6.693 

6.701 

45 

6.708 

6.716 

6.723 

6.731 

6.738 

6.745 

6.753 

6.  760 

6.768 

6-775 

46 

6.782 

6.790 

6-797 

6.804 

6.812 

6.819 

6.826 

6.834 

6.841 

6.848 

47 

6.856 

6.863 

6.870 

6.877 

6.885 

6.892 

6.899 

6.907 

6.914 

6.921 

48 

6.928 

6.935 

6.943 

6.950 

6.957 

6.964 

6.971 

6.979 

6.986 

6.993 

49 

7.000 

7.007 

7.014 

7.021 

7.029 

7.036 

7.043 

7.050 

7.057 

7.064 

50 

7.071 

7.078 

7-085 

7.092 

7.099 

7.106 

7.113 

7. 120 

7.127 

7-134 

51 

7. 141 

7.148 

7-155 

7.  162 

7.169 

7.176 

7.183 

7.190 

7.197 

7.204 

52 

7.  211 

7.218 

7-225 

7.232 

7.239 

7.246 

7.253 

7-259 

7.266 

7-273 

53 

7.280 

7.287 

7-294 

7 -301 

7.308 

7.314 

7.321 

7-328 

7.335 

7-342 

54 

n 

7-348 

7-355 

7.362 

7-369 

7.376 

7.382 

7.389 

7-396 

7 .  403 

7.409 

.0         .1           .2           .3           .4          -5           -6           .7         -8           .9 

Arithmetical  Tables 
Numbers  from  1.00  to  99.9 


19 


n 
SS 

.0          .1          .2          .3          -4          -5          -6          .7          -8          .9 

7.416 

7-423 

7-430 

7-436 

7-443 

7-450 

7-457 

7-463 

7-470 

7-477 

56 

7-483 

7-490 

7-497 

7-503 

7-510 

7-517 

7-523 

7-530 

7-537 

7-543 

57 

7-550 

7-556 

7-563 

7-570 

7-576 

7-583 

7-589 

7-596 

7.603 

7.609 

58 

7.616 

7.622 

7.629 

7-635 

7.642 

7.649 

7-655 

7.662 

7.668 

7-675 

59 

7.681 

7.688 

7-694 

7.701 

7-707 

7-714 

7.720 

7-727 

7-733 

7-740 

6o 

7-746 

7-752 

7-759 

7-765 

7-772 

7-778 

7.785 

7.791 

7-797 

7.804 

6i 

7.810 

7.817 

7-823 

7-829 

7-836 

7-842 

7.849 

7-855 

7.861 

7.868 

62 

7.874 

7.880 

7-887 

7-893 

7-899 

7.906 

7.912 

7.918 

7-925 

7-931 

63 

7-937 

7-944 

7-95° 

7-956 

7.962 

7.969 

7-975 

7-981 

7.987 

7-994 

64 

8.000 

8.006 

8.012 

8.019 

8.025 

8.031 

8-037 

8.044 

8.050 

8.056 

65 

8.062 

8.068 

8.075 

8.081 

8.087 

8.093 

8.099 

8.106 

8. 112 

8. 118 

66 

8.124 

8.130 

8.136 

8.142 

8.149 

8.155 

8.161 

8.167 

8-173 

8.179 

67 

8.185 

8. 191 

8.198 

8.204 

8.210 

8.216 

8.222 

8.228 

8.234 

8.  240 

68 

8.246 

8.252 

8.258 

8.264 

8.270 

8.276 

8.283 

8.289 

8.29s 

8.301 

69 

8.307 

8-313 

8-319 

8.325 

8.331 

8.337 

8.343 

8.349 

8-355 

8.361 

70 

8.367 

8-373 

8-379 

8-385 

8.390 

8.396 

8.402 

8.408 

8.414 

8.420 

71 

8.426 

8.432 

8.438 

8.444 

8.450 

8.456 

8.462 

8.468 

8.473 

8.479 

72 

8.485 

8.491 

8.497 

8.503 

8.509 

8.515 

8.521 

8.526 

8-532 

8-538 

73 

8.544 

8-550 

8.556 

8.562 

8.567 

8.573 

8.579 

8-585 

8.591 

8.597 

74 

8.602 

8.608 

8.614 

8.620 

8.626 

8.631 

8.637 

8.643 

8.649 

8.654 

75 

8.660 

8.666 

8.672 

8.678 

8.683 

8.689 

8.695 

8.701 

8.706 

8.712 

76 

8.718 

8.724 

8.729 

8.735 

8.741 

8.746 

8.752 

8.758 

8.764 

8.769 

77 

8.775 

8.781 

8.786 

8.792 

8.798 

8.803 

8.809 

S.815 

8.820 

8.826 

78 

8.832 

8.837 

8.843 

8.849 

8.854 

8.860 

8.866 

8. 871 

8.877 

8.883 

79 

8.888 

8.894 

8.899 

8.905 

8.911 

8.916 

8.922 

8.927 

8.933 

8-939 

80 

8.944 

8.950 

8.955 

8.961 

8.967 

8.972 

8.978 

8.983 

8.989 

8.994 

81 

9.000 

9.006 

9.011 

9.017 

9.022 

9.028 

9-033 

9-039 

9.044 

9.050 

82 

9-055 

9.061 

9.066 

9.072 

9-077 

9.083 

9.088 

9-094 

9.099 

9-105 

83 

9.  no 

9. 116 

9. 121 

9.127 

9.132 

9.138 

9-143 

9.149 

9-154 

9. 160 

84 

9-165 

9. 171 

9.176 

9.182 

9.187 

9.192 

9.198 

9.203 

9.  209 

9.214 

85 

9.  220 

9.225 

9.230 

9.236 

9.241 

9-247 

9.252 

9-257 

9-263 

9.268 

86 

9.274 

9.279 

9-284 

9.  290 

9-295 

9.301 

9-306 

9-3II 

9-317 

9.322 

87 

9-327 

9-333 

9-338 

9-343 

9-349 

9-354 

9-359 

9-365 

9-370 

9-375 

88 

9-381 

9.386 

9-391 

9-397 

9.402 

9.407 

9-413 

9.418 

9-423 

9.429 

89 

9-434 

9-439 

9-445 

9-450 

9-455 

9.460 

9.466 

9-471 

9-476 

9.482 

90 

9-487 

9-492 

9-497 

9-503 

9-508 

9-513 

9-518 

9-524 

9-529 

9-534 

91 

9-539 

9-545 

9-550 

9-555 

9.560 

9.566 

9-571 

9-576 

9-581 

9-586 

92 

9-592 

9-597 

9.602 

9.607 

9.612 

9.618 

9-623 

9.628 

9-633 

9-638 

93 

9-644 

9-649 

9654 

9-659 

9.664 

9.670 

9-675 

9.680 

9.685 

9.690 

94 

9-695 

9.701 

9.706 

9. 711 

9.716 

9.721 

9.726 

9-731 

9-737 

9.742 

95 

9-747 

9-752 

9-757 

9.762 

9-767 

9.772 

9-778 

9-783 

9-788 

9.793 

96 

9-798 

9.803 

9.808 

9-813 

9.818 

9.823 

9.829 

9-834 

9-839 

9.844 

97 

9.849 

9-854 

9-859 

9.864 

9.869 

9-874 

9.879 

9.884 

9.889 

9.894 

98 

9.899 

9-905 

9.910 

9-915 

9.920 

9-925 

9-930 

9-935 

9.940 

9.945 

99 

n 

9-950 

9-955 

9.960 

9-965 

9.970 

9-975 

9. 980 

9-985 

9-990 

9-995 

.0          .1           .2          .3          .4          -S          .6          .7         -8          .9 

20 


Arithmetical  Tables 


9.  Cuoes  of  Niun- 


n 

012345678             9 

I.O 

1 .000 

1.030 

1. 061 

I  093 

1. 125 

1. 158 

1. 191 

1.225 

1. 260 

1.295 

I.I 

1-331 

1.368 

1.40=; 

1.443 

1.482 

1.521 

1.561 

1.602 

1.643 

1-685 

1.2 

1.728 

1.772 

1. 8x6 

1. 861 

1.907 

1-953 

2.000 

2.048 

2.097 

2.147 

1-3 

2.197 

2.248 

2.300 

2.353 

2.406 

2.460 

2.515 

2-571 

2.628 

2.686 

1.4 

2.744 

2.803 

2.863 

2.924 

2.9S6 

3-049 

3. 112 

3-177 

3-242 

3-308 

i-S 

3-375 

3-443 

3.512 

3.582 

3.- 652 

3.724 

3-796 

3-870 

3-944 

4.020 

1.6 

4.096 

4.173 

4.252 

4.331 

4. 411 

4.492 

4-574 

4.657 

4.742 

4.827 

1-7 

4.913 

5.000 

5.088 

5-178 

5.268 

5-359 

5-452 

5.545 

5.640 

5-735 

1.8 

S-832 

5-93° 

6.029 

6.128 

6.230 

6.332 

6.435 

6.539 

6.645 

6-751 

1-9 

6.859 

6.968 

7.078 

7.189 

7-3°i 

7-415 

7-530 

7.645 

7.762 

7.881 

2.0 

8.000 

8.  121 

8.242 

8.365 

8.490 

8.615 

8.742 

8.870 

8-999 

9.129 

2.1 

9.  261 

9-394 

9-528 

9.664 

9.800 

9-938 

10.08 

10.  22 

10.36 

10.50 

2.2 

10.65 

10.79 

10.94 

II  .09 

11.24 

11-39 

11.54 

11.70 

11.85 

12.01 

2.3 

12. 17 

12-33 

12.49 

12.65 

12.81 

12.98 

13-14 

^3-3T^ 

13.48 

13-65 

2.4 

13.82 

14.00 

14.17 

14.35 

14.53 

14.71 

14.89 

15.07 

15-25 

15-44 

2.5 

15.62 

15.81 

16.00 

16. 19 

16-39 

16.58 

16.78 

16.97 

17.17 

17-37 

2.6 

17-58 

17.78 

17.98 

18.19 

18.40 

18.61 

18.82 

19-03 

19-25 

19-47 

2.7 

19.68 

19.90 

20. 12 

20.35 

20.57 

20.80 

21.02 

21.25 

21. 48 

21.72 

2.8 

21-95 

22. 19 

22.43 

22.67 

22.91 

23.15 

23-39 

23-64 

23-89 

24.14 

2.9 

24.39 

24.64 

24.90 

25-15 

25.41 

25.67 

25.93 

26.20 

26.46 

26.73 

3.0 

27.00 

27-27 

27.54 

27.82 

28.09 

28.37 

28.65 

28.93 

29.22 

29.50 

3-1 

29.79 

30.08 

30.37 

30.66 

30.96 

31 .  26 

31.55 

31.86 

32.16 

32.46 

3-2 

32-77 

33-08 

33-39 

33-70 

34-01 

34.33 

34.65 

34.97 

35.29 

35.61 

3-3 

35-94 

36.26 

36.59 

36.93 

37.26 

37.60 

37.93 

38.27 

38.61 

38.96 

3.4 

39-30 

39-65 

40.00 

40.35 

40.71 

41.06 

41.42 

41.78 

42.14 

42.51 

35 

42.88 

43.24 

43-61 

43.99 

44.36 

4-4.74 

45-12 

45.50 

45.88 

46.27 

3.6 

46.66 

47-05 

47-44 

47.83 

48.23 

48.63 

49-03 

49.43 

49.84 

50.24 

3.7 

50-65 

51.06 

51-48 

51-90 

52.31 

52.73 

53-16 

53.58 

54.01 

54.44 

3.8 

54.87 

55-31 

55-74 

56.18 

56.62 

57-07 

57.51 

57-96 

58.41 

58.86 

3-9 

59-32 

59-78 

60.24 

60.70 

61.16 

61.63 

62. 10 

62.57 

63.04 

63.52 

4.0 

64.00 

64.48 

64.96 

65.45 

65.94 

66.43 

66.92 

67-42 

67.92 

68.42 

4.1 

68. 92 

69-43 

69-93 

70.44 

70.96 

71.47 

71-99 

72-51 

73-03 

73.56 

4.2 

74.09 

74-62 

75-15 

75-69 

76.23 

76.77 

77-31 

77-85 

78.40 

78.95 

4-3 

79-51 

80.06 

80.62 

81.18 

81.75 

82.31 

82. 88 

83-45 

84-03 

84.60 

4.4 

85.18 

85-77 

86. 35 

86.94 

87.53 

88.12 

88.72 

89-31 

89-92 

90.52 

4-5 

gi.  12 

91-73 

92.35 

92.96 

93-58 

94.20 

94.82 

95-44 

96.07 

96.70 

4.6 

97-34 

97-97 

98.61 

99.25 

99.90 

100.5 

loi.  2 

101.8 

102.  5 

1.03 . 2 

4.7 

103.8 

104.5 

105.  2 

105 . 8 

106.  5 

107.  2 

107.9 

108.5 

109.2 

109.9 

4.8 

no. 6 

III 

3 

1 12.0 

112. 7 

113. 4 

114. 1 

114.8 

"5-5 

116.2 

116. 9 

4-9 

117. 6 

118 

4 

1 19. 1 

119.8 

120.6 

121.3 

122.0 

122.8 

123.5 

124.3 

S.o 

125.0 

125 

8 

126.5 

127-3 

128.0 

128.8 

129.6 

130.3 

131-1 

131-9 

5-1 

132-7 

133 

4 

134.2 

135-0 

135-8 

136.6 

137-4 

138.2 

139-0 

139.8 

5-2 

140.6 

141 

4 

142.  2 

I43-I 

143-9 

144.7 

145-5 

146.4 

147-2 

148.0 

5-3 

148.9 

149 

7 

150.6 

151-4 

152-3 

I53-I 

154-0 

154-9 

155-7 

156.6 

5-4 

I57-S 

158 

3 

159.2 

160. 1 

161. 0 

161. 9 

162.8 

163.7 

164.6 

165.5 

;( 

012345678            9 

Arithmetical  Tables 
bers  from  1.00  to  9.99 


21 


n 

012345678            9 

5-5 

166.4 

167.3 

168.2 

169. 1 

170.0 

171. 0 

171. 9 

172.8 

173-7,174-7 

5.6 

175-6 

176.6 

177-5 

178.5 

179-4 

180.4 

181. 3 

182.3 

183-3 

184.2 

5-7 

185.2 

186.2 

187. 1 

188. 1 

iSg.i 

190. 1 

191 . 1 

192. 1 

193- 1 

194. 1 

5.8 

195. 1 

196. 1 

197. 1 

198.2 

199.2 

200.  2 

201 . 2 

202.3 

203.3 

204.3 

5-9 

205.4 

206.4 

207-5 

208.5 

209.6 

210.6 

211 . 7 

212.8 

213.8 

214.9 

6.0 

216.0 

217. 1 

218.2 

219.3 

220.3 

221.4 

222.5 

223.6 

224.8 

225.9 

6.1 

227.0 

228.1 

229.2 

230.3 

231-5 

232.6 

233 -7 

234-9 

236.0 

237-2 

6.2 

238.3 

239-5' 

240.6 

241.8 

243.0 

244.1 

245-3 

246.5 

247.7 

248.9 

6.3 

250.0 

251.2 

252-4 

253-6 

254.8 

2!;6.o 

257-3 

258.5 

259-7 

260.9 

6.4 

262. 1 

263.4 

264.6 

265.8 

267.  I 

268.3 

269.6 

270.8 

272.1 

273-4 

6.5 

274.6 

275.9 

277.2 

278.4 

279-7 

281.0 

282.3 

283.6 

284.9 

286.2 

6.6 

287.5 

288.8 

290. 1 

291.4 

292.8 

294.1 

295-4 

296.7 

298.1 

299.4 

6.7 

300.8 

302.1 

303-5 

304.8 

306.2 

307-5 

308.9 

310.3 

311-7 

313-0 

6.8 

314.4 

315.8 

317-2 

318.6 

320.0 

321-4 

322.8 

324-2 

325-7 

327-1 

6.9 

328.5 

329-9 

331-4 

332.8 

334.3 

335-7 

337-2 

338.6 

340.1 

341-5 

7.0 

343 -o 

344.5 

345-9 

347-4 

348.9 

350-4 

351-9 

353-4 

354.9 

356-4 

7.1 

357-9 

359-4 

360.9 

362.5 

364.0 

365-5 

367-1 

368.6 

370.1 

371-7 

7.2 

373-2 

374-8 

376.4 

377-9 

379-5 

381. 1 

382.7 

384-2 

385-8 

387-4 

7.3 

389.0 

390.6 

392.2 

393-8 

395-4 

397-1 

398.7 

400.3 

401.9 

403.6 

7.4 

405.2 

406.9 

408.5 

410.2 

411-S 

413-S 

415-2 

416.8 

418.5 

420.2 

7.5 

421.9 

423.6 

425-3 

427.0 

428.7 

430-4 

432-1 

433-8 

435.5 

437-2 

7.6 

439-0 

440.7 

442.5 

444-2 

445-9 

447-7 

449-5 

451-2 

453-0 

454-8 

7.7 

456-5 

458.3 

460. 1 

461.9 

463-7 

465-5 

467-3 

469.1 

470.9 

472-7 

7.8 

474-6 

476-4 

478.2 

4S0.0 

481.9 

483-7 

485.6 

487.4 

489.3 

491-2 

7.9 

493-0 

494-9 

496.8 

498-7 

500.6 

502.5 

504-4 

506.3 

508.2 

510. 1 

8.0 

512.0 

513-9 

515.8 

517-8 

519-7 

521-7 

523-6 

525-6 

527-S 

529-5 

8.1 

531.4 

533-4 

535-4 

537-4 

539-4 

541-3 

543-3 

545-3 

547-3 

549-4 

8.2 

551-4 

553-4 

555-4 

557-4 

559-5 

56I-S 

563-6 

565-6 

567-7 

569-7 

8.3 

571-8 

573-9 

575-9 

578.0 

580.1 

582.2 

584-3 

586.4 

588.5 

590.6 

8.4 

592.7 

594-8 

596-9 

599-1 

601 . 2 

603.4 

605-5 

607.6 

609.8 

612.0 

8.5 

614. 1 

616.3 

618.5 

620.7 

622.8 

625.0 

627.2 

629.4 

631.6 

633-8 

8.6 

636.1 

638.3 

640.5 

642.7 

645-0 

647.2 

649-5 

651.7 

654.0 

656.2 

8.7 

658.5 

660.  8 

663.1 

665-3 

667.6 

669. 9 

672.2 

674.5 

676.8 

679.2 

8.8 

681.5 

683.8 

686.1 

688.5 

690.8 

693.2 

695-5 

697.9 

700.2 

702.6 

8.9 

705.0 

707.3 

709.7 

712. 1 

714-5 

716.9 

719-3 

721.7 

724-2 

726.6 

9.0 

729.0 

731-4 

733-9 

736.3 

738-8 

741-2 

743-7 

746.1 

748.6 

751-1 

9.1 

753.6 

756.1 

758-6 

761.0 

763-6 

766.1 

768.6 

771. 1 

773-6 

776.2 

9.2 

778.7 

781.2 

783.8 

786.3 

788.9 

791-5 

794.0 

796.6 

799.2 

801.8 

9-3 

804.4 

807.0 

809.6 

812.2 

814.8 

817-4 

820.0 

822.7 

825.3 

827-9 

9.4 

830.6 

833.2 

835-9 

838.6 

841.2 

843-9 

846.6 

849-3 

852.0 

854-7 

9-5 

857-4 

860.1 

862.8 

865.5 

868.3 

871.0 

873-7 

876.5 

879-2 

882.0 

9.6 

884.7 

887.5 

890-3 

893-1 

895-8 

898.6 

901.4 

904.2 

907.0 

909-9 

9-7 

912.7 

915-5 

918.3 

921 . 2 

924.0 

926.9 

929-7 

932-6 

935-4 

938.3 

9.8 

941.2 

944-1 

947.0 

949-9 

952-8 

955-7 

958.6 

961-5 

964.4 

967.4 

9.9 

970.3 

973-2 

976.2 

979-1 

982.1 

985-1 

988.0 

991.0 

994.0 

997-0 

n 

012345678            9 

22 


Arithmetical  Tables 
10.  Cube  Roots  of  Numbers 


n 

</n 

V  io» 

11 

</n 

"V  lo  n 

■y  100  n 

'\Jioo  n 

10 

2.1544 

4.6416 

10.000 

55 

3-8030 

8.1932 

17.652 

II 

2. 2240 

4-7914 

10.323 

56 

3-8259 

8.2426 

17-758 

12 

2. 2894 

49324 

10. 627 

57 

3-8485 

8.2913 

17.863 

13 

2.3513 

5-0658 

10. 914 

S8 

3-8709 

8.3396 

17.967 

14 

2.4101 

5-1925 

II. 187 

59 

3-8930 

8.3872 

18.070 

IS 

2. 4662 

5-3133 

11.447 

60 

3-9149 

8.4343 

18.171 

i6 

2.5198 

5.4288 

11.696 

61 

3-9365 

8 . 4809 

18.272 

17 

2-5713 

5-5397 

11-935 

62 

3-9579 

8.5270 

18.371 

i8 

2. 6207 

5.6462 

12. 164 

63 

3-9791 

8.5726 

18.469 

19 

2.6684 

5-7489 

12.386 

64 

4. 0000 

8.6177 

18.566 

20 

2.7144 

5.8480 

12-599 

6S 

4. 0207 

8.6624 

18.663 

21 

2.7589 

5  -  9439 

12.806 

66 

4.0412 

6. 7066 

18.758 

22 

2.8020 

6. 0368 

13.006 

67 

4.0615 

8.7503 

18.852 

23 

2.8439 

6.  1269 

13.200 

68 

4.0817 

8.7937 

18.945 

24 

2.88/J5 

6.2145 

13-389 

69 

4. 1016 

8.8366 

19.038 

25 

2.9240 

6.2996 

13-572 

70 

4.1213 

8. 8790 

19.129 

36 

2.9625 

6.3825 

13-751 

71 

4. 140S 

8.9211 

19.220 

27 

3 . 0000 

6.4633 

13-925 

72 

4. 1602 

8.9628 

19.310 

28 

3-0366 

6.5421 

14.095 

73 

4.1793 

9.0041 

19.399 

29 

3-0723 

6.6191 

14.260 

74 

4.1983 

9-0450 

19.487 

30 

3.1072 

6.6943 

14.422 

75 

4.2172 

9-0856 

19.574 

31 

3-1414 

6.7679 

14.581 

76 

4-2358 

9.1258 

19.661 

32 

3-1748 

6.8399 

14.736 

77 

4-2543 

9-1657 

19.747 

33 

3-2075 

6. 9104 

14.888 

78 

4.2727 

9.2052 

19.832 

34 

3-2396 

6-9795 

15-037 

79 

4. 2908 

9-2443 

19.916 

35 

3.2711 

7-0473 

15-183 

80 

4.3089 

9.2832 

20.000 

36 

3-3019 

7.1138 

15-326 

81 

4-3267 

9-3217 

20.083 

37 

33322 

7.1791 

15-467 

82 

4-3445 

9-3599 

20. 165 

38 

3.3620 

7-2432 

15-605 

83 

4.3621 

9-3978 

20.247 

39 

3-3912 

7.3061 

15-741 

84 

4-3795 

9-4354 

20.328 

40 

3 . 4200 

7.3681 

15-874 

8s 

4.3968 

94727 

20.408 

41 

3-4482 

7.4290 

16.005 

86 

4.4140 

9-5097 

20.4S8 

42 

3-4760 

7.4889 

16.134 

87 

4-4310 

9  5464 

20.567 

43 

3-5034 

7-5478 

16. 261 

88 

4-4480 

9.5828 

20.646 

44 

3-5303 

7-6059 

16.3S6 

89 

4-4647 

9. 6190 

20.724 

4S 

3-5569 

7-6631 

16.510 

90 

4.4814 

9.6549 

20,  801 

46 

3-5830 

7-7194 

16. 631 

91 

4.4979 

9 ■ 6905 

20.878 

47 

3.6088 

7-7750 

16.751 

92 

4-5144 

9-7^59 

20.954 

48 

3  6342 

7.8297 

16.869 

93 

4-5307 

9.7610 

21 .029 

49 

3-6593 

7-8837 

16.985 

94 

4.5468 

9-7959 

21. 105 

50 

3 ■ 6840 

7-9370 

17. 100 

9S 

4-5629 

9-8305 

21.179 

51 

3-7084 

7 . 9896 

17.2:3 

96 

4-5789 

9.8648 

21-253 

52 

3-7325 

8. 0415 

17-325 

97 

4-5947 

9 . S990 

21.327 

S3 

3  7563 

8.0927 

17-435 

98 

4 . 6104 

9.9329 

2 1 . 400 

54 

3- 7798 

8-1433 

17-544 

99 

4.6261 

9.9666 

21.472 

Arithiietical  Tables 
11.  Three-Halves  Powers  of  Numbers 


23 


n 

01            23456             789 

o.o 

0.000 

O.OOI 

0.003 

0.005 

0.008 

O.OII 

0.015 

0.019 

0.023 

0.027 

O.I 

0.032 

0.036 

0.042 

0.047 

0.052 

0.058 

0.  064 

0. 070 

0.076 

0.083 

0.2 

0.089 

0.096 

0.103 

0.  no 

0.118 

0. 125 

0-133 

0. 140 

0. 148 

0.  156 

0.3 

0. 164 

0-173 

0.181 

0. 190 

0. 198 

0.  207 

0.  216 

0.  225 

0.234 

0.244 

0.4 

0-253 

0.263 

0.272 

0.  282 

0.  292 

0.302 

0.312 

0.322 

0-333 

0.343 

0.5 

°-354 

0.364 

0-375 

0.386 

0.397 

0.408 

0.419 

0.430 

0.442 

0-453 

0.6 

0.465 

0.476 

0.488 

0.500 

0.512 

0.524 

0.536 

0.548 

0.561 

0-573 

0.7 

0.586 

0.598 

0.611 

0.624 

0.637 

0.650 

0.663 

0.676 

0.689 

0.702 

0.8 

0.716 

0.729 

0.743 

0.756 

0.770 

0.784 

0.798 

0.811 

0.826 

0.840 

0.9 

0.854 

0.868 

0.882 

0.897 

0.911 

0.926 

0.941 

0-955 

0.970 

0-985 

1.0 

1. 000 

1.015 

1.030 

1.045 

1.061 

1.076 

1.091 

1 .  107 

1 .  122 

1. 138 

I.I 

I-IS4 

1. 170 

1-185 

1. 201 

1.  217 

1-233 

1.249 

1.266 

1.282 

1.298 

1.2 

1-315 

I-33I 

1-348 

1.364 

1.381 

1-398 

1.414 

I -43 1 

1.448 

1-465 

1-3 

1.482 

1.499 

1-S17 

1-534 

I -551 

1-569 

1.586 

1.604 

1. 621 

1-639 

1.4 

1.657 

1.674 

1.692 

1 .  710 

1.728 

1.746 

1.764 

1.782 

1.800 

1.819 

1-5 

1-837 

1.856 

1.874 

1-893 

1. 911 

1-930 

1.948 

1.967 

1.986 

2.005 

1.6 

2.024 

2.043 

2.062 

2.081 

2. 100 

2. 119 

2.139 

2.158 

2.178 

2-197 

1.7 

2.217 

2.236 

2.256 

2.275 

2  -  295 

2-315 

2.335 

2-355 

2.37s 

2-395 

1.8 

2.415 

2-435 

2-455 

2.476 

2.496 

2.516 

2.537 

2-557 

2-578 

2-598 

1.9 

2.619 

2.640 

2.660 

2.681 

2.  702 

2.723 

2-744 

2.765 

2.786 

2.807 

2.0 

2.828 

2.850 

2.871 

2.892 

2.914 

2-935 

2-957 

2.978 

3.000 

3.021 

2.1 

3 -043 

3-065 

3-087 

3.109 

3-131 

3-153 

3-175 

3-197 

3.219 

3-241 

2.2 

3   263 

3-285 

3-308 

3-330 

3-353 

3-375 

3-398 

3-420 

3.443 

3-465 

2.3 

3.488 

3-5" 

3-534 

3-557 

3-580 

3-602 

3.626 

3-649 

3.672 

3-695 

2.4 

3-718 

3-741 

3-765 

3-788 

3. 811 

3-835 

3-858 

3-882 

3 .  906 

3-929 

2.5 

3-953 

3-977 

4.000 

4.024 

4.048 

4.072 

4.096 

4. 120 

4.144 

4.168 

2.6 

4.192 

4-217 

4.241 

4.263 

4.289 

4.314 

4.338 

4-363 

4.387 

4-412 

2.7 

4-437 

4.461 

4.486 

4-511 

4-536 

4.560 

4.585 

4.  610 

4-635 

4.660 

2.8 

4.685 

4.710 

4.736 

4-761 

4.786 

4.811 

4-837 

4.862 

4.888 

4-913 

2.9 

4-939 

4.964 

4.990 

5-015 

5-041 

5.067 

5-093 

5-118 

5-144 

5-170 

3-0 

5-196 

5.222 

5-248 

5-274 

5-300 

5-327 

5-353 

5.379 

5-405 

5-432 

3-1 

5-458 

5-485 

5-5II 

5-538 

5-564 

5-591 

5-617 

5-644 

5-671 

5.698 

3-2 

5-724 

5-751 

5.778 

5-805 

5-832 

5-859 

5-886 

5-913 

5  •  940 

5-968 

3-3 

5-995 

6.022 

6.049 

6.077 

6. 104 

6.132 

6-159 

6.186 

6.214 

6.242 

3-4 

6.  269 

6.297 

6.325 

6-352 

6.380 

6.408 

6.436 

6.464 

6.492 

6.520 

3.5 

6-548 

6.576 

6.604 

6.632 

6.660 

6.689 

6.717 

6.745 

6-774 

6.802 

3.6 

6.831 

6.859 

6.888 

6.916 

6.945 

6.973 

7.002 

7-031 

7-059 

7.088 

3-7 

7. 117 

7-146 

7-175 

7.204 

7-233 

7.262 

7.291 

7-320 

7-349 

7-378 

3-8 

7.408 

7-437 

7.466 

7-495 

7-525 

7-554 

7-584 

7-613 

7-643 

7.672 

3-9 

7.702 

7-732 

7.761 

7.791 

7.821 

7-850 

7.880 

7.910 

7.940 

7-970 

4.0 

8.000 

8.030 

8.060 

8.090 

8.120 

8.150 

8'  181 

8.211 

8.241 

8.272 

4.1 

8.302 

8-332 

8.363 

8.393 

8.424 

8.454 

8.485 

8-515 

8.546 

8-577 

4.2 

8.607 

8.638 

8.669 

8.  700 

8-731 

8.762 

8.793 

8.824 

8-855 

8.886 

4-3 

8.917 

8.948 

8-979 

9.010 

9.041 

9-073 

9.104 

9-135 

9.167 

9.198 

4.4 

9.230 

9.261 

9-293 

9-324 

9-356 

9-387 

9-419 

9-451 

9-482 

9-514 

n 

01            23456789 

24  Arithmetical  Tables 

12.  Fifth  Powers  and  Roots;  Five-Halves  Powers  and  Roots 


n 

O.I 

n' 

»* 

n^ 

n^ 

n 
4.6 

w" 

n* 

«s 

n« 

0.0000 

0.6310 

0.0032 

0.3981 

2059. 

6  1.3569 

45-383 

I. 8412 

0.2 

0 . 0003 

0. 7248 

0.0179 

0.5253 

4-7 

2293. 

5  1.3628 

47.890 

1-8571 

0.3 

0.0024 

0.7860 

0.0493 

0.6178 

4.8 

2548. 

0  1.3685 

50.47S 

1.8728 

0.4 

0.0102 

0.8326 

0. 1612 

0.6931 

4-9 

2824. 

8  1-3742 

53-148 

1 . 8883 

0.5 

0.0312 

0.8706 

0.1768 

0.7579 

5-0 

3125. 

0  1.3797 

55-902 

1-9037 

0.6 

0.0778 

0. 9029 

0. 2789 

0.8152 

S.I 

3450. 

3  1.3852 

58.739 

1.9188 

0.7 

0.1681 

0.9311 

0.4100 

0.8670 

5.2 

3802. 

0  1.3906 

61.661 

1-9338 

0.8 

0.3277 

0.9564 

0.5724 

0.9146 

5-3 

4182. 

0  1.3959 

64.668 

I  -  9485 

0.9 

0-5905 

0.9791 

0.7684 

0.9587 

5.4 

4591- 

7  1. 401 1 

67.762 

1.9632 

I.O 

I . 0000 

I . 0000 

I . 0000 

1 . 0000 

5-5 

5032. 

8  1.4063 

70.943 

1.9776 

I.I 

I. 6105 

I .0192 

I  2691 

1.0389 

5.6 

5507- 

3  1-4114 

74.211 

1-9919 

1.2 

2 . 4883 

1.0371 

1-5774 

1-0757 

5-7 

6016. 

9  I. 4164 

77.569 

2.0061 

1.3 

3-7129 

1-0539 

1 .9269 

I. 1107 

5.8 

6563- 

6  I. 4213 

81.016 

2.0201 

1.4 

5-3782 

I . 0696 

2.3191 

1.1441 

5-9 

7149- 

2  1.4262 

84.553 

2.0340 

i.S 

7-5938 

1.0845 

2-7557 

I. 1761 

6.0 

7776. 

3  1.4310 

88.182 

2.0477 

1.6 

10.486 

1.0986 

3.2382 

1.2068 

6.1 

8446. 

3  1.4357 

91 . 902 

2.0613 

1.7 

14.199 

I. 1120 

3.7681 

1-2365 

6.2 

9161. 

3  I -4404 

95.715 

2.0747 

1.8 

18.896 

1.1247 

4.3469 

1.2651 

6.3 

9924-' 

%   1-4450 

99.621 

2.0880 

1-9 

24.761 

1-1370 

4.9760 

1.2927 

6.4 

10737 

1.4496 

103.62 

2. 1012 

2.0 

32.000 

1.14S7 

5-6569 

1-3195 

6.5 

11603 

1-4541 

107.72 

2.1143 

2.1 

40. 841 

1 . 1600 

6.3907 

1-3455 

6.6 

12523 

1-4585 

III  ^ I 

2. 1272 

2.2 

Si-536 

1.1708 

7.1789 

1.3708 

6.7 

13501 

I . 4629 

116. 19 

2.1401 

2.3 

64-363 

1.1813 

8.0227 

I -3954 

6.8 

14539 

1 . 4672 

120. 58 

2. 1528 

2.4 

79.626 

1.1914 

8-9234 

1-4193 

6.9 

15640 

I-471S 

125.06 

2.1654 

2.5 

97-656 

I . 201 1 

9.8821 

1.4427 

7.0 

16807 

1-4758 

129.64 

2.1779 

2.6 

118. 81 

1 . 2106 

10.900 

1-4655 

7-1 

1S042 

I  4800 

134.32 

2.1903 

2.7 

143-49 

I. 2198 

11.979 

1.4878 

7.2 

19349 

1.4841 

139-10 

2. 2026 

2.8 

172. 10 

I. 2287 

13. 119 

1.5096 

7-3 

20731 

1.4882 

143-98 

2.2148 

2.9 

205.11 

1-2373 

14.322 

1-5309 

7.4 

22190 

-  1-4923 

148.96 

2.2269 

3-0 

243. 00 

1-2457 

15-588 

1-5518 

7.5 

23730 

1-4963 

154.05 

2.2388 

3-1 

286.29 

1-2539 

16.920 

1-5723 

7.6 

25355 

1 . 5002 

159.23 

2.2507 

3-2 

335-54 

I. 2619 

18.318 

1-5924 

7-7 

27068 

1-5042 

164.52 

2.2625 

3-3 

391-35 

1.2697 

19-783 

1.6122 

7.8 

28872 

1.5081 

169.92 

2.2742 

3-4 

454-35 

1-2773 

21.316 

1-6315 

7.9 

30771 

I-5119 

175.42 

2.2859 

3-S 

525.22 

1.2847 

22.918 

1-6505 

8.0 

32768 

I-5157 

181.02 

2.2974 

3.6 

604 . 66 

1.2920 

24-590 

I .6692 

8.2 

37074 

1.5232 

192.55 

2.3202 

3.7 

693-44 

I . 2991 

26.333 

1.6876 

8.4 

41821 

1-5306 

204.50 

2.3427 

3.8 

7.92-35 

1.3060 

28. 149 

1-7057 

8.6 

47043 

1-5738 

216.89 

2.3648 

3.9 

902. 24 

I. 3128 

30.037 

1.7236 

8.8 

52773 

1.5449 

229. 72 

2.3S67 

4.0 

1024.0 

1-3195 

32.000 

1.7411 

9.0 

59049 

1-5518 

243.00 

2.4082 

4.1 

1158.6 

1.3260 

34-038 

1-7584 

9.2 

65908 

1-5587 

256.73 

2.4295 

4.2 

1306.9 

1-3324 

36.151 

1-7754 

9.4 

73390 

1  -  5654 

270.91 

2-4505 

4-3 

1470. I 

1-3387 

38.342 

1.7922 

9.6 

81537 

1.5720 

285.55 

2.4712 

4.4 

1649. 2 

1-3449 

40. 610 

I . 8088 

9.8 

90392 

1-5785 

300.65 

2.4917 

4-5 

1845-3 

1-3510 

42.957 

1.8251 

10 

lOOOOC 

J  1.5849 

316.23 

2.5119 

Explanation  on  page  39 


Arithmetical  Tables  25 

13.  Explanations 

All  of  the  Arithmetical  Tables,  except  10  and  12,  are  four- 
place  tables,  that  is,  the  values  of  the  functions  are  given  to  four 
significant  figures.  In  Tables  10  and  12  five  significant  figures  are 
given.  For  all  these  tables  the  probable  error  in  the  last  figure 
is  one-fourth  of  a  unit. 

Table  G  gives  Reciprocals  of  all  numbers  having  three  sig- 
nificant figures  by  properly  moving  the  decimal  point.  Thus 
the  reciprocals  of  0.705,  7.05,  705,  and  0.0705  are  1.418,  0.1418, 
0.001418,  and  14.18  to  four  significant  figures. 

Table  7  gives  Squares  of  all  niunbers  having"  three  significant 
figures  by  properly  moving  the  decimal  point.  Thus  the  squares 
of  3.94,  0.394,  and  39.4  are  15.52,  0.1552,  and  1552  to  four  sig- 
nificant figures.  Here  the  decimal  point  moves  two  places  in  the 
function  when  it  moves  one  place  in  the  argument. 

Table  8  gives  Square  Roots  of  all  numbers  of  three  significant 
figures.  For  the  numbers  5.42  and  542  the  square  roots  2.328 
and  23.28  are  found  on  page  16;  for  the  numbers  54.2  and  5420  the 
scjuare  roots  7.302  and  73.62  are  found  on  page  18.  Here,  as  in 
Table  7,  the  decimal  point  moves  two  places  in  the  square 
when  it  moves  one  place  in  the  square  root. 

Table  9  gives  Cubes  of  all  numbers  of  three  significant  figures  by 
moving  the  decimal  point  three  places  in  the  function  when  it 
moves  one  place  in  the  argument.  Thus,  the  cubes  of  4.69  and 
0.469  are  103.2  and  0.1032  to  four  significant  figures. 

Table  10  gives  Cube  Roots  of  two-place  numbers.  Thus,  the 
cube  root  of  35  is  3.2711,  that  of  350  is  7.0473,  and  that  of  3500 
is  15.183.  When  the  given  number  contains  a  decimal  point, 
multiply  it  by  1000,  take  the  root  from  the  table  and  then  divide 
this  by  10.  In  this  manner  the  cube  roots  of  3.5,  0.35,  and  0.0035 
are  found  to  be  1.5183,  0.7047,  and  0.3271. 

Table  11  gives  values  of  ?i^  or  V  n'  for  values  of  n  up  to  4.49. 
This  is  useful  in  the  weir  computations  of  hydraulics.  For  ex- 
ample, when  n  is  2.59  the  value  of  n^  is  5.432.  Table  12  is  also 
used  in  hydraulic  work  in  computations  on  the  flow  of  water  in 
long  pipes. 


26  Arithmetical  Tables 

14.  Exercises  for  Students 

1.  Find  the  reciprocals  of  0.14,  0.145,  and  0.1456;    also  of  1.4, 
1.45,  and  1.456;  also  of  14,  14.5,  and  14.56. 

2.  Find  the  reciprocals  of  0.90,  0.99,  and  0.909;    also  of  0.695, 
0.6954,  and  6.954;  also  of  0.295,  0.2954,  29.5,  and  295.4. 

3.  Find  the  squares  of  3.90,  3.902,  and  3.909;    also  of  39.0,  39.02, 
and  39.09;  also  of  0.77,  0.777,  and  0.7777;  also  of  0.707  and  0.7071. 

4.  Find  the  square  roots  of  the  following  numbers: 

2.08  2.081  2.087 

2.09  2.091  2.097 
9.43                           9.433                         9.435 

20.8  20.81  20.87 

20.9  20.91  20.97 
94.3                            94.33  94.35 

0.89  0.891  0.8913 

89.0  89.1  89.13 

8900  8910  8913 

5.  Find  the  value  of  V3-+4-  +  122  by  the  help  of  Tables  7  and  8. 

6.  Find  the  values  of  the  following  functions: 

4.232  =  4.2312=  42.312  = 

8.952=  89.52=  8952  = 

0.7232=  0.07232=  7232  = 

7.253=  72.5'=  725'  = 

7.043=  0.7043=  7043  = 

0.9993  9  993  99.93  = 

7.  Find  the  value  of  \/¥+A^+5'^  by  the  help  of  Table  8. 

8.  Find  the  three-halves  powers  of  2.78  and  2.783. 

9.  Find  the  values  of  the  following  functions: 
1/32.2     =.  10/9.8     =  20/0.45 
\/64r32   =                 V19I6         =  VO  1916   = 
1.35*       =                    1.35*       =  1.35i     =■  . 


Chapter  3 
TABLES  OF  CIRCLES  AND  SPHERES 


28 


Circles  and  Spheres 

15.  Areas  of  Circles  for  Diam- 


d 

0123456789 

I.O 

0.785 

0.801 

0.817 

0.833 

0.  849 

0.866 

0.882 

0.  899 

0.916 

0-933 

I.I 

0.950 

0.968 

0.985 

1 .003 

1. 021 

1-039 

1-057 

1-075 

1.094 

1. 112 

1.2 

1. 131 

1 .  150 

1. 169 

1. 188 

1.208 

1.227 

1.247 

1 .  267 

1.287 

1-307 

1.3 

1.327 

1-348 

1-368 

1-389 

1 .  410 

1-431 

1-453 

1-474 

1-496 

1-517 

1.4 

1.539 

I.  561 

1.584 

1 .  606 

1.629 

1-651 

1-674 

1.697 

1.720 

1-744 

1. 5 

1.767 

I. 791 

1.815 

1-839 

1.863 

1.8S7 

1 .  91 1 

1.936 

1.961 

1.986 

1.6 

2. on 

2-036 

2.061 

2.087 

2. 112 

2.138 

2. 164 

2. 190 

2.217 

2.243 

1.7 

2.270 

2-297 

2-324 

2-351 

2-378 

2.405 

2-433 

2.461 

2.488 

2-516 

1.8 

2.545 

2-573 

2.602 

2-630 

2-659 

2.688 

2.717 

2.746 

2.776 

2.806 

1.9 

2.83s 

2.865 

2.895 

2.926 

2.956 

2.986 

3-017 

3.048 

3-079 

3. no 

2.0 

3.142 

3-173 

3-205 

3.237 

3.269 

3-301 

Z2Z3 

3.365 

3-398 

3-431 

2.1 

3.464 

3-497 

3530 

3.563 

3-597 

3-631 

3.664 

3.698 

3-733 

3-767 

2.2 

3.801 

3-836 

3.871 

3-906 

3  941 

3-976 

4.012 

4-047 

4-083 

4.119 

2.3 

4.155 

4. 191 

4.227 

4-264 

4-301 

4.337 

4-374 

4-412 

4-449 

4-486 

2.4 

4-524 

4.562 

4.600 

4.638 

4-676 

4.714 

4-753 

4-792 

4-831 

4-870 

2.5 

4.909 

4-948 

4.988 

5.027 

3-067 

5.107 

5-147 

5-187 

5.228 

5.269 

2.6 

5-309 

5-350 

5-391 

S-433 

5-474 

S-51S 

5-557 

5-599 

5.641 

5 -683 

2.7 

5-726 

5.768 

5.811 

5-853 

5.896 

5.940 

5-983 

6.026 

6.070 

6. 114 

2.8 

6.158 

6.  202 

6.246 

6.  290 

6-335 

6-379 

6.424 

6.469 

6.514 

6.560 

2.9 

6-605 

6.651 

6.697 

6-743 

6-789 

6-835 

6.881 

6.928 

6-975 

7.022 

3.0 

7.069 

7. 116 

7.163 

7. 211 

7-258 

7-306 

7-354 

7.402 

7-451 

7-499 

3-1 

7.548 

7.596 

7.64s 

7.694 

7-744 

7-793 

7-843 

7-892 

7-942 

7.992 

3-2 

8.042 

8.093 

8.143 

8.194 

8-245 

8.296 

8.347 

8-398 

8-450 

8.501 

3-3 

8.553 

8.605 

8.657 

8.709 

8.  762 

8.814 

8.867 

8.920 

8-973 

9.026 

3.4 

9.079 

9-133 

9.186 

9.240 

9-294 

9-348 

9  402 

9-457 

9-5" 

9.566 

3-5 

9.621 

9.676 

9.731 

9.787 

9.842 

9.898 

9-954 

10.01 

10.07 

10. 12 

3.6 

10. 18 

10.  24 

10.29 

10.35 

10.41 

10.46 

10.  52 

10.58 

10.64 

10.69 

3.7 

10.75 

10.81 

10.87 

10.93 

10.99 

11.04 

11. 10 

11. 16 

11.22 

n.28 

3.8 

1 1  •  34 

11.40 

11.46 

11.52 

11-58 

11.64 

11.70 

n.76 

11.82 

11.88 

3-9 

11.95 

12.01 

12.07 

12.13 

12. 19 

12.25 

12.32 

12.38 

12-44 

12.50 

4.0 

12.57 

12.63 

12.69 

12.76 

12.82 

12.88 

12.95 

13.01 

13-07 

13-M 

4.1 

13.20 

13-27 

13.33 

13.40 

13-46 

13-53 

13-59 

13-66 

13-72 

13-79 

4.2 

13.85 

13-92 

13.99 

14.05 

14. 12 

14-19 

14.25 

14-32 

14-39 

14-45 

4-3 

14-52 

14.59 

14.66 

14.73 

14.79 

14-86 

14.93 

15.00 

15-07 

15-14 

4.4 

15.  21 

15-27 

15.34 

15.41 

15-48 

15-55 

15.62 

15-69 

15-76 

15-83 

4.5 

15-90 

15-98 

16.05 

16. 12 

16. 19 

16.26 

16.33 

16.40 

16.47 

i6-5S 

4.6 

16.62 

16.69 

i6.7fe 

16.84 

16.91 

16.98 

17.06 

17-13 

17.  20 

17.28 

4.7 

17-35 

17-42 

17-50 

17.57 

17-65 

17-72 

17.80 

17.87 

17-95 

18.02 

4.8 

18.  10 

18.17 

18.25 

18.32 

18.40 

18.47 

18-55 

18.63 

18.70 

18.78 

4-9 

i8.86 

18.93 

19.01 

19.09 

19.17 

19-24 

19.32 

19-40 

19.48 

19-56 

S.o 

19-63 

19.71 

19-79 

19.87 

19-95 

20.03 

20. 11 

20. 19 

20.27 

20.35 

51 

20.43 

20.51 

20.59 

20.67 

20- 75 

20.83 

20.91 

20.99 

21.07 

21. 16 

S.2 

21.  24 

21.32 

21.40 

21.48 

21-57 

21.65 

21-73 

21. 81 

21.90 

21.98 

5.3 

22.06 

22. 15 

22.23 

22.31 

22.40 

22.48 

22.56 

22.65 

22.73 

22.82 

5.4 

22.90 

22.99 

23-07 

23-16 

23-24 

23-33 

23-41 

23-50 

23-59 

23-67 

d 

012345678            9 

Circles  and  Spheres 
eters  in  Units  and  Hundredths 


29 


d 

0 

I           2 

3 

4 

56789 

s-s 

23.76 

23 

.84 

23^93 

24 

.  02 

24 

.  1 1 

24 

•19 

24.28 

24^37 

24-45 

24.54 

5.6 

24-63 

24 

•72 

24.81 

24 

.89 

24 

.98 

25 

•07 

25. 16 

25^25 

25-34 

25^43 

5.7 

25-52 

25 

.61 

25.70 

25 

■79 

25 

.88 

25 

.97 

26.06 

26. 15 

26.  24 

26.33 

5.8 

26.42 

26 

-51 

26.60 

26 

.69 

26 

•79 

26 

.88 

26.97 

27.06 

27-15 

27.25 

5-9 

27 -.34 

27 

•43 

27^53 

27 

.62 

27 

•71 

27 

.81 

27.90 

27.99 

28.09 

28.18 

6.0 

28.27 

28 

•37 

28.46 

28 

•56 

28 

•65 

28 

•75 

28.84 

28.94 

29-03 

29.13 

6.1 

29.  22 

29 

•32 

29.42 

29 

•51 

29 

.61 

29 

•71 

29.80 

29.90 

30.00 

30.09 

6.2 

30-19 

30 

.29 

30-39 

30 

•  48 

30 

•58 

30 

.68 

30-78 

30.88 

30-97 

31.07 

6.3 

31-17 

31 

.27 

31-37 

31 

•47 

31 

•57 

31 

.67 

31-77 

31-87 

31-97 

32.07 

6.4 

32.17 

32 

.27 

32.37 

32 

•47 

32 

•57 

32 

.67 

32.78 

32.88 

32.98 

33.08 

6.5 

33.18 

Z3 

■29 

33-39 

33, 

•49 

33 

■59 

33 

.70 

33-80 

33.90 

34.00 

34-11 

6.6 

34.21 

34 

32 

34.42 

34 

■52 

34 

•63 

34 

•73 

34-84 

34.94 

35-05 

35.15 

6.7 

35-26 

35 

36 

35-47 

35 

•57 

35 

68 

35 

.78 

35-89 

36.00 

36. 10 

36.21 

6.8 

36-32 

36 

42 

36-53 

36 

64 

36 

75 

36.85 

36.96 

37.07 

37-18 

37-28 

6.9 

37-39 

37 

50 

37-61 

37 

72 

37 

83 

37 

94 

38.05 

38.16 

38.26 

38.37 

7.0 

38.48 

38 

59 

38-70 

38 

82 

38 

93 

39 

04 

39^15 

39.26 

39-37 

39-48 

7.1 

39-59 

39 

70 

39-82 

39 

93 

40 

04 

40 

15 

40.26 

40.38 

40.49 

40.60 

7.2 

40.72 

40 

83 

40.94 

41 

06 

41 

17 

41 

28 

41.40 

41.51 

41.62 

41.74 

7.3 

41.85 

41 

97 

42.08 

42 

20 

42 

31 

42 

43 

42.54 

42.66 

42.78 

42.89 

7.4 

43-01 

43 

12 

43-24 

43 

36 

43 

47 

43 

59 

43-71 

43-83 

43.94 

44.06 

7.5 

44.18 

44 

30 

44.41 

44 

53 

44 

65 

44 

77 

44-89 

45^01 

45-13 

45.25 

7.6 

45-36 

45 

48 

45  •  60 

45 

72 

45 

84 

45 

96 

46.08 

46.20 

46.32 

46.45 

7.7 

46.57 

46.69 

46.81 

46 

93 

47 

05 

47 

17 

47-29 

47^42 

47-54 

47.66 

7.8 

47-78 

47 

91 

48.03 

48 

15 

48 

27 

48 

40 

48.52 

48.65 

48.77 

48.89 

7.9 

49.02 

49 

14 

49.27 

49 

39 

49 

51 

49 

64 

49-76 

49.89 

50.01 

50-14 

8.0 

50.27 

50. 

39 

50.52 

50 

64 

50 

77 

50 

90 

51.02 

51.15 

51.28 

51-40 

8.1 

51-53 

51- 

66 

51^78 

51 

91 

52 

04 

52 

17 

52.30 

52^42 

52-55 

52.68 

8.2 

52.81 

52. 

94 

53^07 

53 

20 

53 

33 

S3 

46 

53-59 

53^72 

53-85 

53^98 

8.3 

54.11 

54- 

24 

54-37 

54 

50 

54. 

63 

54 

76 

54.89 

55^02 

55-15 

55^29 

8.4 

55-42 

55- 

55 

55-68 

55 

81 

55. 

95 

56. 

08 

56.21 

56^35 

56.48 

56.61 

8.5 

56-75 

56. 

88 

57.0I 

57- 

15 

57- 

28 

57. 

41 

57.55 

57-68 

57-82 

57^95 

8.6 

58.09 

58. 

22 

58.36 

58- 

49 

58- 

63 

58. 

77 

58.90 

59.04 

59-17 

59^31 

8.7 

59-45 

59- 

58 

59-72 

59- 

86 

59- 

99 

60. 

13 

60.  27 

60.41 

60-55 

60.68 

8.8 

60.82 

60. 

96 

61. 10 

61. 

24 

61. 

38 

61. 

51 

61.65 

61.79 

61-93 

62.07 

8.9 

62.21 

62. 

35 

62.49 

62. 

63 

62. 

77 

62. 

91 

63-05 

63-19 

63-33 

63.48 

9.0 

63.62 

63- 

76 

63.90 

64- 

04 

64  • 

18 

64. 

33 

64.47 

64.61 

64-75 

64.90 

9.1 

65.04 

65- 

18 

65-33 

65- 

47 

65^ 

61 

65. 

76 

65.90 

66.04 

66. 19 

66.33 

9.2 

66.48 

66. 

62 

66.77 

66. 

91 

67. 

06 

67. 

20 

67-35 

67.49 

67-64 

67.78 

9-3 

67-93 

68. 

08 

68.22 

68 

37 

68. 

51 

68. 

66 

68.81 

68.96 

69. 10 

69.25 

9.4 

69.40 

69. 

55 

69.69 

69 

84 

69. 

99 

70. 

14 

70-29 

70.44 

70-58 

70.73 

9.5 

70.88 

71. 

03 

71.18 

71- 

2,3 

71. 

48 

71- 

63 

71.78 

71-93 

72.08 

72.23 

9.6 

72.38 

72. 

53 

72.68 

72 

84 

72. 

99 

73. 

14 

73^29 

73.44 

73-59 

73-75 

9.7 

73-9° 

74- 

OS 

74-20 

74 

36 

74. 

51 

74. 

66 

74.82 

74-97 

75-12 

7=;-28 

9.8 

75-43 

75- 

58 

75-74 

75- 

89 

76. 

05 

76. 

20 

76.36 

76.51 

76.67 

76.82 

9.9 

76.98 

77- 

13 

77.29 

77^ 

44 

77^ 

60 

77^ 

76 

77.91 

78.07 

78.23 

78-38 

d 

0            I 

2            3 

456789 

30 


Circles  akd  Spheres 
16.  Areas  of  Circles 


Diameters  in  Units 

and  Eiithtb 

s 

d 

o 

0                 1/8             1/4                3/8               1/2               5/8              3/4              Vs 

0 . 0000 

0.0123 

0.0491 

0. 1104 

0.1963 

0.3068 

0.4418 

0.6013 

I 

0-7854 

0. 9940 

1.2272 

I . 4849 

1.7671 

2.0739 

2.4053 

2.7612 

2 

3-1416 

3-5466 

3.9761 

4.4301 

4.9087 

5.4119 

5.9396 

6.4918 

3 

7.0686 

7.6699 

8.2958 

8.9462 

9.6211 

10.321 

1 1 . 045 

11-793 

4 

12.566 

13-364 

14.186 

15.033 

15-904 

16.800 

17.721 

18.665 

5 

19-635 

20.629 

21.648 

22.691 

23.758 

24.850 

25.967 

27.109 

6 

28.274 

29465 

30. 680 

31-919 

33.183 

34472 

35.785 

37.122 

7 

38-485 

39-871 

41 . 282 

42.718 

44.179 

45.664 

47.173 

43.707 

8 

50-265 

51.849 

53-456 

55-088 

56.745 

58.426 

60.132 

61.862 

9 

63.617 

65.397 

67. 201 

69.029 

70.882 

72.760 

74.662 

76-589 

10 

78-540 

80.516 

82.516 

84-541 

86.590 

88. 664 

90.763 

92.886 

II 

95  033 

97.205 

99-402 

lOI . 62 

103.87 

106. 14 

108.43 

110.75 

13 

113. 10 

115.47 

117.86 

120. 28 

122. 72 

125.19 

127.68 

130.19 

13 

132-73 

135.30 

137-89 

140.50 

143     14 

145.80 

148.49 

151.20 

14 

153-94 

156.70 

159.48 

162.30 

165.13 

167.99 

170.87 

173.78 

IS 

176.71 

179.67 

182.65 

185.66 

188.69 

191-75 

194.83 

197.93 

i6 

201.06 

204. 22 

207.39 

210. 60 

213-82 

217.08 

220.35 

223.65 

17 

226.98 

230-33 

233.71 

237.10 

240.53 

243-98 

247.45 

250.9s 

i8 

254-47 

258.02 

261.59 

265.18 

268.80 

272.45 

276. 12 

279.81 

19 

283.53 

287.27 

291.04 

294.83 

298.65 

302.49 

306.35 

310.24 

20 

314.16 

318.10 

322.06 

326.05 

330.06 

334.10 

338.16 

342.25 

21 

346.36 

350.50 

354-66 

358.84 

363-05 

367.28 

371.54 

375.83 

22 

380.13 

384.46 

388.82 

393 . 20 

397-61 

402.04 

406 . 49 

410.97 

23 

415-48 

420.00 

424-56 

429.13 

433-74 

438.36 

443-01 

447.69 

24 

452-39 

457.11 

461.86 

466.64 

471-44 

476.26 

481. II 

485.98 

25 

490.87 

495.79 

500.74 

505-71 

510.71 

515.72 

520.77 

525.84 

26 

530.93 

536.05 

541.19 

546.35 

551-55 

556.76 

562.00 

567.27 

27 

572.56 

577.87 

583.21 

588.57 

593 . 96 

599.37 

604. 81 

610. 27 

28 

615.75 

621 . 26 

626.80 

632-36 

637-94 

643.55 

649.18 

654.84 

29 

660.52 

666.23 

671.96 

677.71 

683.49 

689.30 

695-13 

700.98 

30 

706.86 

712.76 

718.69 

724.64 

730.62 

736.62 

742.64 

74S.69 

31 

754.77 

760.87 

766.99 

773-14 

779.31 

785.51 

791-73 

797.98 

32 

804. 25 

810. 54 

8:6.86 

823.21 

829.58 

835-97 

842.39 

848. S3 

33 

855-30 

861.79 

868.31 

874-85 

881.41 

S8S.00 

894.62 

901 . 26 

34 

907.92 

914.61 

921.32 

928.06 

934.82 

941.61 

948.42 

955.25 

35 

962. II 

969.00 

975.91 

982.84 

989.80 

996.78 

1003.8 

1010.8 

36 

1017.9 

1025 . 0 

1032.  I 

1039. 2 

1046.3 

1053-5 

1060. 7 

1068.0 

37 

1075.2 

1082. 5 

1089.8 

1097-1 

1104.5 

1 1 1 1 .  8 

1 1 19. 2 

ii'26.  7 

38 

"34-1 

II4I.6 

1149.1 

1 156. 6 

1164.2 

1171.7 

1179.3 

1186.9 

39 

1194.6 

1202.3 

1210.0 

1217.7 

1225.4 

1233.2 

1241.0 

1 248 . 8 

40 

1256.6 

1264.5 

1272.4 

1280.3 

1288.2 

1296. 2 

1304.2 

1312.2 

41 

1320.3 

1328.3 

1336.4 

1344.5 

1352.7 

1360.8 

1369-0 

1377.2 

42 

1385.4 

1393.7 

1402.0 

1410.3 

1418.6 

1427.0 

1435-4 

1443.8 

43 

1452.2 

1460. 7 

1469. 1 

1477.6 

14S6.2 

1494.7 

1503.3 

1511.9 

44 
d 

1520.5 

1529.2 

1537 -9 

1546.6     1555.3 

1564.0 

1572-8 

1581.6 

0                1/8              1/4                3/8               1/2               6/8               3/4             7/8 

Circles  and  Spheres 
17.  Circumferences  of  Circles 


31 


E 

iameters 

in  Units 

and  Tenths 

d 

.0           .1           .2          .3           .4          .5           .6           .7           .8          .9 

0 

o.  ooo 

0.314 

0.628 

0.942 

i-257li-57i    1-885   2.199 

2.513 

2.827 

I 

3-142 

3-456 

3-770 

4.084 

4.398  4.712  5.02715.341 

5-655 

5-969 

2 

6.283 

6-597 

6.912 

7.226    7.540    7.85418.168:8.482 

8.796 

9.  Ill 

3 

9425 

9-739 

10.05 

10.37 

10.68   11.00 

II. 31 

11.62 

11.94 

12.  25 

4 

12.57 

12.88 

13-19 

13-51 

13.82  14.14 

14.45 

14.77 

15-08 

15-39 

5 

15-71 

16.02 

16.34 

16.65 

16.96  17.28 

17-59 

17.91 

18.22 

18.54 

6 

18.85 

19. 16 

19.48 

19-79 

20.  II 

20.42    20.73 

21.05 

21.36 

21.68 

7 

21.99 

22.31 

22.62 

22.93 

23.25 

23.56    23.88 

24.19 

24-50 

24.82 

8 

25-13 

25-45 

25-76 

26.08 

26.39 

26.70 

27.02 

27-33 

27-65 

27.96 

9 

28.27 

28.59 

28.90 

29.  22 

29-53 

29-85 

30.16 

30-47 

30-79 

31.10 

lO 

31-42 

31-73 

32-04 

32.36 

32.67 

32-99 

33.30 

33-62 

33-93 

34-24 

II 

34-56 

34.87 

35-19 

35-50 

35-81 

36-13 

36.44 

36.76 

37-07 

37.38 

12 

37-70 

38-01 

38-33 

38.64 

38-96 

39.27    39.58 

39-90 

40. 21 

40.53 

13 

40.84 

41-15 

41.47 

41.78 

42. 10 

42.41 

42.73 

43-04 

43.35 

43-67 

14 

43-98 

44  30 

44.61 

44.92 

45-24 

45-55 

45-87 

46.18 

46.50 

46.81 

15 

47.12 

47-44 

47-75 

48.07 

48.38 

48.69 

49.01 

49.32 

49.64 

49-95 

i6 

50-27 

50.58I 50.89 

51  .  21 

51-52 

51-84 

52.15 

52.46 

52.78 

53-09 

17 

53-41 

53-72 

54-04 

54.35   54-66 

54-98 

55-29 

55 -61 

55-92 

56-23 

iS 

56-55 

56.86 

57.18 

57.49I57.81 

58.12 

58.43 158.75 

59.06 

59-38 

19 

59-69 

60.00   60.32 

60.63   60.95 

61.26 

61. 58  61.89 

62 .  20 

62.  52 

18.   Circumferences  of  Circles 


Diameters  in  Units  and  Eighth 

s 

d 

0 

0                  1/8                1/4                3/8                1/2                5/8                3/4               7/8 

0. 0000 

0.3927 

0.7854 

1.17S1 

1.5708 

1-9635 

2.3562 

2.7489 

I 

3-1416 

3-5343 

3.9270 

4-3197 

4.7124 

5-1051 

5-4978 

5-8905 

2 

6.2832 

6.6759 

7.0686 

7-4613 

7-8540 

8.2467 

8.6394 

9.0321 

3 

9.4248 

9-8175 

10. 210 

10.603 

10. 996 

11.388 

II. 781 

12.174 

4 

12.566 

12-959 

13-352 

13-744 

14-137 

14-530 

14.923 

15-315 

5 

15.708 

16. lOI 

16.493 

16.886 

17.279 

17.671 

18.064 

18-457 

6 

18.850 

19.242 

19.635 

20.028 

20.420 

20.813 

21.206 

21.598 

7 

21 . 991 

22.384 

22.777 

23.169 

23.562 

23-955 

24-347 

24.740 

8 

25-133 

25   525 

25.918 

26.311 

26. 704 

27.096 

27-489 

27.882 

9 

28.274 

28.667 

29.060 

29-452 

29-S45 

30-23S 

30-631 

31.023 

10 

31-416 

31.809 

32.201 

32-594 

32.987 

?>3  ■  379 

33-772 

34.165 

II 

34-558 

34-950 

35-343 

35-736 

36.128 

36-521 

36-914 

37  306 

12 

37-699 

38-092 

38-485 

38-877 

39-270 

39  663 

40-055 

40.448 

13 

40.841 

41.233 

41 . 626 

42.019 

42.412 

42. 804 

43-197 

43-590 

14 

43-982 

44-375 

44.768 

45. 160 

45-553 

45-946 

46-338 

46.731 

IS 

47.124 

47-517 

47-909 

48.302 

48-695 

49.087 

49.480 

49-873 

16 

50-265 

50.658 

51-051 

51-444 

51-836 

52.229 

52. 622 

53-014 

17 

53-407 

53 -800 

54-192 

54-585 

54-978 

55-371 

55-763 

56-156 

18 

56-549 

56-941 

57-334 

57-727 

58.119 

58-512 

58-905 

59-298 

19 

59.690 

60.083 

60.476 

60.868 

61.261 

61.654 

62.046 

62.439 

32 


Circles  axd  Spheres 


19.  Circular 


Central 

Length 

Rise 

Area 

Central 

Length 

Rise 

Area 

Angle 

of 

of 

of 

Angle 

of 

of 

of 

Degrees 

Chord 

Arc 

Segment 

.Degrees 

Chord 

Arc 

Segment 

I 

0.0175 

0 . 0000 

0.00000 

46 

0.7815 

0.0795 

0.04176 

3 

0-0349 

0.0002 

0. 00000 

47 

0.7975 

0.0829 

0.04448 

3 

0.0524 

0 . 0003 

0. OOOOI 

48 

0.8135 

0.0865 

0.04731 

4 

0.0698 

0. 0006 

0.00003 

49 

0.8294 

0.0900 

0.05025 

5 

0.0872 

0. 0010 

0 . 00006 

50 

0.8452 

0.0937 

0.05331 

6 

0.1047 

0.0014 

O.OOOIO 

51 

0. 8610 

0.0974 

0.05649 

7 

0.  1221 

0. 0019 

0. 00015 

52 

0.8767 

0. 1012 

0.05978 

8 

0.139s 

0.0024 

0. 00023 

53 

0.8924 

0. 1051 

0.06319 

9 

0. 1569 

0.0031 

0. 00032 

54 

0. 9080 

0. 1090 

0.06673 

10 

0.1743 

0. 0038 

0 . 00044 

55 

0.9235 

0.1130 

0.07039 

n 

O.I9I7 

0.0046 

0.00059 

56 

0.9389 

0. 1171 

0.07417 

13 

0. 2091 

0.0055 

0.00076 

57 

0.9543 

0. I2I2 

0.07808 

13 

0. 2264 

0. 0064 

0.00097 

58 

0.9696 

0.1254 

0.08212 

14 

0.2437 

0. 0075 

0.00121 

59 

0.9848 

0. 1296 

0.08629 

IS 

0. 2611 

0.0086 

0.00149 

60 

I . 0000 

0.1340 

0.09059 

i6 

0.2783 

0.0097 

0.00181 

61 

1.0151 

0.1384 

0.09502 

17 

0. 2956 

0. 01 10 

0.00217 

62 

I. 0301 

0. 1428 

0.09958 

i8 

0.3129 

0.0123 

0.00257 

63 

1.0450 

0.1474 

0. 10428 

19 

0.3301 

0.0137 

0.00302 

64 

I . 0598 

0. 1520 

0. 10911 

20 

0.3473 

0.0152 

0.00352 

65 

1.0746 

0. 1566 

0. 1 1408 

31 

0.3645 

0.0167 

0 . 00408 

66 

1.0893 

0.1613 

0. 11919 

32 

0.3816 

0. 0184 

0.00468 

67 

I. 1039 

0. 1661 

0.12443 

23 

0.3987 

0.0201 

0.00535 

68 

I. 1 1 84 

0. I7IO 

0. 12982 

24 

0.4I5S 

0. 0219 

0.00607 

69 

I. 1328 

0.1759 

0.1353s 

25 

0.4329 

0.0237 

0.00686 

70 

I. 1472 

0.1808 

0. 14102 

26 

0.4499 

0.0256 

0.00771 

71 

I . 1614 

0. 1859 

0. 14683 

27 

0. 4669 

0.0276 

0. 00862 

72 

1. 1756 

0. I9IO 

•0.15279 

28 

0.4838 

0.0297 

0. 00961 

73 

I. 1896 

0. 1961 

0.15889 

29 

0. 5008 

0.0319 

0.01067 

74 

I. 2036 

0. 2014 

0. 16514 

30 

0. 5176 

0.0341 

0 . 0 1 1 80 

75 

I. 2175 

0. 2066 

0.17154 

31 

O.S34S 

0.0364 

0.01301 

76 

I. 2313 

0. 2120 

0.17808 

32 

0.5312 

0.0387 

0. 01429 

77 

1.2450 

0.2174 

0.18477 

33 

0. 5680 

0. 0412 

0.01566 

78 

I . 2586 

0. 2229 

0. 19160 

34 

0.5847 

0.0437 

0.01711 

79 

I . 2722 

0.2284 

0. 19859 

35 

0. 6014 

0.0463 

0. 01864 

80 

1.2856 

0.2340 

0.20573 

36 

0 . 6 1 80 

0.0489 

0.02027 

81 

I . 2989 

0. 2396 

0.-2I30I 

37 

0. 6346 

0.0517 

0.02198 

82 

I.3121 

0.2453 

0. 22045 

38 

0. 651 1 

0.0545 

0.02378 

83 

1.3252 

0.2510 

0. 22804 

39 

0.6676 

0.0574 

0.02568 

84 

1.3383 

0. 2569 

0.23578 

40 

0.6840 

0.0603 

0.02767 

85 

1.3512 

0. 2627 

0.24367 

41 

0. 7004 

0.0633 

0.02976 

86 

I . 3640 

0. 2686 

0.25171 

42 

0.7167 

0.0664 

0.03195 

87 

1.3767 

0. 2746 

0. 25990 

43 

0.7330 

0.0696 

0.03425 

88 

I . 3893 

0. 2807 

0. 26825 

44 

0.7492 

0.0728 

0.03664 

89 

I. 401 8 

0.2867 

0.2767s 

45 

0.7654 

0. 0761 

0.03915 

90 

1.4142 

0. 2929 

0.  28540 

Segments 


Circles  and  Spheres 


33 


Central      L 

ength 

Rise 

Area 

Central      L 

ength 

Rise 

Area 

Angle 

of 

of 

Angle 

of 

of 

of 

Degrees       "■ 

I^hord 

Arc 

Segment 
0. 29420 

Degrees      C 

-hord 

Arc 

Segment 

91         I 

4265 

0. 2991 

136         I 

.8544 

0. 6254 

0.83949 

92         I 

4387 

0-3053 

0.30316 

137         I 

.8608 

0.633s 

0.85455 

93         I 

4507 

0.3116 

0. 31226 

138         I 

8672 

0.6416 

0.86971 

94         I 

4627 

0.3180 

0.32152 

139         I 

8733 

0.6498 

0.88497 

95         I 

4746 

0.3244 

0.33093 

140         I 

8794 

0. 6580 

0.90034 

96         I 

4863 

0.3309 

0.34050 

141         I 

8853 

0.6662 

0. 91580 

97         1 

4979 

0.3374 

0.35021 

142         I 

8910 

0.6744 

0.93135 

98         I 

5094 

0.3439 

0.36008 

143         I 

8966 

0.6827 

0. 94700 

99        I 

5208 

0.3506 

0.37009 

144         I 

9021 

0. 6910 

0. 96274 

100         I 

532^ 

°-3572 

0. 38026 

145         I 

9074 

0.6993 

0.97858 

lOI             I 

5432 

0.3639 

0.39058 

146         I 

9126 

0.7076 

0.99449 

102         I 

5543 

0.3707 

0.40104 

147         I 

9176 

0. 7160 

I .01050 

103         I 

5652 

0.3775 

0. 41166 

148         I 

9225 

0.7244 

1.02658 

104         I 

5760 

0.3843 

0.42242 

149        I 

9273 

0.7328 

1.04275 

los         I 

5867 

0.3912 

0-43333 

ISO        I 

9319 

0.7412 

1.05900 

106         I 

5973 

0.3982 

0.44439 

iSi         I 

9363 

0.7496 

1-07532 

107         I 

6077 

0.4052 

0. 45560 

152         I 

9406 

0.7581 

1.09171 

108         I 

6180 

0.4122 

0. 46695 

153         I 

9447 

0. 7666 

1.10818 

109         I 

6282 

0.4193 

0.47844 

154         I 

9487 

0.7750 

I. 12472 

no          I 

6383 

0.4264 

0. 49008 

155         I 

9526 

0.7836 

1.14132 

in         I 

6483 

0.4336 

0. 50187 

156         I 

9563 

0. 7921 

I .15799 

112         I 

6581 

0.4408 

0.51379 

157         I 

9598 

0.8006 

I. 17472 

113         I 

6678 

0. 4481 

0.52586 

158         I 

9633 

0. 8092 

1.19151 

114         I 

6773 

0.4SS4 

0.53807 

159         I 

9665 

0.8178 

1.20835 

115         I 

6868 

0.4627 

0.55041 

160         I 

9696 

0.8264 

1.22525 

116         I 

6961 

0.4701 

0.56289 

161         I 

9726 

0.8350 

I . 24221 

117         I 

7053 

0.4775 

0.57551 

162         I 

9754 

0.8436 

I. 25921 

118         I 

7143 

0.4850 

0.58827 

163         I 

9780 

0. 8522 

I. 27626 

119         I 

7233 

0.4925 

0. 601 16 

164         I 

9805 

0.8608 

1-29335 

120         I 

7321 

0. 5000 

0. 61418 

165         I 

9829 

0.8695 

I. 31049 

121         I 

7407 

0. 5076 

0.62734 

166         I 

9851 

0.8781 

1.32766 

122         I 

7492 

0.5152 

0.64063 

167         I 

9871 

0.8868 

1.34487 

123         I 

7576 

0. 5228 

0.65404 

168         I 

9890 

0.8955 

I. 362 1  2 

124         I 

7659 

0-5305 

0.66759 

169        I 

9908 

0. 9042 

1.37940 

125         I 

7740 

0.5383 

0.68125 

170         I 

9924 

0.9128 

I. 39671 

126         I 

7820 

0. 5460 

0.69505 

171         I 

9938 

0.9215 

I. 41404 

127         I 

7899 

0.5538 

0. 70897 

172         I 

9951 

0.9302 

I. 43140 

128         I 

7976 

0. 5616 

0.72301 

173         I 

9963 

0.9390 

1.44878 

129         I 

8052 

0.5695 

0.73716 

174         I 

9973 

0.9477 

I. 46617 

130         I 

8126 

0.5774 

0.75144 

175         I 

9981 

0.9564 

1-48359 

131         I 

8199 

0.5853 

0.76584 

176         1 

9988 

0. 9651 

1.50101 

132         I 

8271 

0-5933 

0. 78034 

177         I 

9993 

0.9738 

I-51845 

133         I 

8341 

0.6013 

0.79497 

178         I 

9997 

0.9825 

1.53589 

134         I 

8410 

0.6093 

0. 80970 

179         I 

9999 

0.9913 

1-55334 

135         I 

8478 

0.6173 

0.82454 

180        2 

0000 

I . 0000 

I. 57080 

34 


Circles  and  Spheres 


20.  Volumes  of  Spheres 

Diameters  in  Units  and  Tenths 


d 

0 

.0    .1     .2    .3     .4     .5    .6    .7     .8    .9 

0.000 

O.OOl 

0.004 

0.014 

0.034 

0.065 

0.113 

0. 180 

0.268 

0.382 

I 

0.524 

0.697 

0.905 

1. 150 

1-437 

1.767 

2.145 

2.572 

3-054 

3-591 

2 

4.189 

4.849 

5-575 

6-371 

7-238 

8.181 

9.203 

10.31 

11.49 

12.77 

3 

14.14 

15.60 

17. 16 

18.82 

20.58 

22.45 

24.43 

26.52 

28.73 

31.06 

4 

33-51 

36.09 

38-79 

41.63 

44.60 

47-71 

50-97 

54-36 

57-91 

61.60 

5 

65-45 

69.46 

73.62 

77-95 

82.45 

87.11 

91-95 

96.97 

102.2 

107-5 

6 

113. 1 

118. 8 

124.8 

130.9 

137-3 

143-8 

150-5 

157-5 

164.6 

172.0 

7 

179.6 

187.4 

195-4 

203.7 

212.  2 

220.9 

229.8 

239.0 

248.5 

258.2 

8 

268.1 

278-3 

288.7 

299.4 

310.3 

321.6 

333 -o 

344.8 

356.8 

369-1 

9 

381-7 

394-6 

407-7 

421 . 2 

434-9 

448.9 

463.2 

477-9 

492.8 

508.0 

lO 

523 -6 

539-5 

555-6 

572.2 

589.0 

606. 1 

623.6 

641.4 

659.6 

678.1 

II 

696.9 

716. 1 

735-6 

755-5 

775-7 

796.3 

817-3 

838-6 

860.3 

882.3 

12 

904.8 

927.6 

950.8 

974-3 

998.3 

1023 

1047 

1073 

1098 

1124 

13 

1 1 50 

1177 

1204 

1232 

1260 

1288 

1317 

1346 

1376 

1406 

14 

1437 

1468 

1499 

1531 

1563 

1596 

1630 

1663 

1697 

1732 

15 

1767 

1803 

1839 

187s 

1912 

1950 

1988 

2026 

2065 

2105 

i6 

2145 

2185 

2226 

2268 

2310 

2352 

2395 

2439 

2483 

2527 

17 

2572 

2618 

2664 

2711 

2758 

2806 

285  s 

2903 

2953 

3003 

i8 

3054 

3i°S 

3157 

3209 

3262 

3315 

3369 

3424 

3479 

3535 

19 

3.S9I 

3648 

3706 

3764 

3823 

3882 

3942 

4003  4064 

4126 

21.  Volumes  of  Spheres 

Diameters  in  Units  and  Eighths 


d 

0 

0 

1/8 

1/4 

3/8 

1/2 

5/8 

3/4    7/8 

0.0000 

O.OOIO 

0.0082 

0.0276 

0.0654 

O.127S 

0.2209 

0.3508 

I 

0.5236 

0.74S5 

1.0227 

1.3612 

1.7671 

2.2468 

2 . 8062 

3-451S 

2 

4.1888 

5.0243 

S-9641 

7-0144 

8.1812 

9.4708 

10.S89 

12-443 

3 

14-137 

15-979 

17.974 

20.129 

22.449 

24.942 

27.612 

30.466 

4 

33.510 

36.751 

40.194 

43.846 

47.713 

51.800 

56.115 

60.663 

5 

65-450 

70.482 

75-766 

81.308 

87.114 

93.189 

99-541 

106.17 

6 

113-10 

120.31 

127.83 

135.66 

143.79 

152.25 

161.03 

170.14 

7 

179-59 

189.39 

199-53 

210.03 

220.89 

232.12 

243-73 

255-71 

8 

268.08 

280.85 

294.01 

307. 58 

321.56 

335-95 

350.77 

366.02 

9 

381.70 

397-83 

414.40 

431.43 

448.92 

466.88 

485-30 

504.21 

10 

523-60 

543-48 

563.86 

584.74 

606.13 

628.04 

650.47 

673-42 

11 

696.91 

720.94 

745-51 

770.64 

796.33 

82 2. 58 

849.40 

876.80 

12 

904-78 

933-35 

962.51 

992.28 

1022.7 

1053-6 

1085.2 

1117-S 

13 

1150.3 

1183.8 

1218.0 

1252.8 

1288.2 

1324-4 

1361.2 

1398.6 

14 

1436.8 

1475.6 

151S-1 

1555.3 

1596.3 

1637-9 

1680.3 

1723-3 

15 

1767.1 

1811.7 

1857.0 

1903.0 

1949.8 

1997.4 

204s -7 

2094.8 

16 

2144.7 

2195-3 

2246.8 

2299.0 

2352.1 

2405.9 

2460.6 

2516. I 

17 

2572.4 

2629.6 

2687.6 

2746. s 

2806.2 

2866.7 

2928.2 

2990. 5 

18 

3053-6 

3117-7 

3182.6 

3248. 5 

331S-2 

3382.9 

3451.5 

3520.9 

19 

3591.4 

3662.7 

3735-0 

3808.2 

38S2.4 

3957.6 

4033-7 

4110.7 

Circles  and  Spheres 
22.  Multipliers  for  Finding  Lengths  of  Circular  Arcs 


35 


I 

2 
3 

4 
5 
6 

7 
8 
9 

Degrees 

Minutes 

Seconds 

0.017453293 
0.034906585 
0.052359878 

0.069813170 
0.087266463 
0.104719755 

0. 122173048 
0.139626340 
0.157079633 

0.00029088S 
0.000581776 
0.000872665 

0.001163553 

0.001454441 
0.001745329 

0.002036217 
0.002327106 
0.002617994 

0.000004848 
0 . 000009696 
0.000014544 

0.000019393 
0.000024241 
0.000029089 

0.000033937 
0.000038785 
0.000043633 

Example. 
Find  length  of  arc  for  a  central 
angle  of  48°  4/  in  circle  of 
12  ft.  radius. 

40°     0.698132 

8°       .139626 

40'       .011636 

5'       .001454 

0.85085 
12 

Length  =  10. 210  ft 

23.  Explanations 

Table  15  gives  Areas  of  Circles  to  four  places  for  three-place 
diameters.  Since  the  area  of  a  circle  varies  as  the  square  of  its 
diameter,  it  follows  that  the  decimal  point  moves  two  places  in 
the  function  when  it  moves  one  place  in  the  argument.  Thus, 
for  diameters  of  4.53  and  45.3  inches  the  areas  of  the  circles  are 
16.12  and  1612  square  inches;  for  a  diameter  of  0.453  inches  the 
area  is  0.1612  square  inches. 

Table  16  gives  Areas  of  Circles  when  the  diameters  are  ex- 
pressed in  imits  and  eighths-;  thus  for  a  diameter  of  22|  inches, 
the  area  is  393.20  square  inches.  When  the  diameter  is  given  to 
sixteenths  the  area  is  approximately  half-way  between  the  two 
nearest  tabular  values;  thus,  for  a  diameter  of  2^6  inches  the  area 
is  3.34  square  inches. 

Tables  17  and  IS  give  Circumferences  of  Circles  for  diameters 
in  tenths  and  eighths  of  units.  For  example,  circles  of  7.2  and  7$ 
inches  in  diameter  have  circumferences  of  22.62  and  22.78  inches. 

Tables  17-18  can  also  be  used  for  finding  a  diameter  when 
the  area  or  circumference  is  given.  Examples:  when  the  areas 
50.52  and  51.34  are  given  the  corresponding  diameters  are  8.02 
and  8.085;  when  the  circumferences  5.027  and  5.134  are  given,  the 
diameters  are  1.600  and  1.634. 

Table  19  gives  properties  of  Segments  of  a  Circle  of  radius 
unity.     For  any  other  radius  r  the  tabular  lengths  of  chord  and 


36  Circles  and  Spheres 

rise  of  arc  are  to  be  multiplied  by  r  and  the  tabular  area  by  r^ 
For  example,  when  the  radius  is  20  feet  and  the  angle  at  the  cen- 
ter of  the  circle  is  82°,  the  length  of  the  chord  of  the  segment  is 
26.242  feet,  the  rise  of  the  arc  is  4.906  feet,  and  the  area  of  the 

segment  is  88.18  square  feet. 

Tables  20  and  21  give  Volumes  of  Spheres  for  diameters  in 
tenths  and  eighths.  Thus,  for  spheres  9.1  and  9|  inches  in  diam- 
eter the  volumes  are  394.G  and  397.8  cubic  inches. 

Table  22  gives  Multipliers  for  finding  lengths  of  Circular  Arcs 
of  radius  unity.  Example:  to  find  the  length  of  a  railroad  curve 
of  700  feet  radius  and  60°  8'  central  angle;  here  the  table  gives 
1.0472  for  C0°  and  0.0023  for  8';  adding  these  and  multiplying  by 
700  gives  734.65  feet  for  the  actual  length  of  the  curve. 

24.  Exercises 

1.  Find  the  areas  of  circles  whose  diameters  are  3.4,  3.42,  and 
3.421  feet;  also  for  diameters  of  340,  342,  and  342.1  feet. 

2.  Find  the  area  for  a  circle  of  19.25  inches  dian.cter  by  inter- 
polation in  Table  15  and  comjjare  the  result  with  that  given  in  Table 
16. 

3.  Find  circumferences  of  circles  20.3  and  2.03  inches  diameter; 
also  of  circles  40.6  and  4.06  feet  diameter. 

4.  In  a  circle  of  12  inches  diameter  the  measured  chord  of  a  seg- 
ment was  14.44  inches.  What  is  the  chord  for  a  radius  unity? 
By  help  of  Table  19  find  the  central  angle,  the  rise  of  the  arc,  and  the 
area  of  the  segment. 

5.  For  a  central  angle  of  48°  30'  find  the  length  of  chord,  rise  of 
arc,  and  area  of  segment  in  a  circle  whose  radius  is  60.5  centimeters. 

6.  What  are  the  volumes  of  spheres  of  0.34,  3.4,  and  34  inches? 

7.  A  cannon  ball  8  inches  in  diameter  has  a  specific  gravity  of  7.8. 
If  the  weight  of  a  cubic  foot  of  water  is  62.5  pounds,  what  is  the  weight 
of  the  cannon  ball? 

8.  Find  the  length  of  a  railroad  curve  having  a  central  angle  of 
3°  15'  and  a  radius  of  5730  feet. 


Chapter  4 
NATURAL  TRIGONOMETRIC  FUNCTIONS 


38 


Trigonometric  Functions 


25.  Nahiral  Sines 


/' 

SINE 

Angl( 

;   0'     10'    20'    30'     40'    50'     60' 

89 

o" 

0.00000 

0.00291 

0.00582 

0.00873 

0. 01164 

0.01454 

0.01745 

I 

0.01745 

0.02036 

0.02327 

0.02618 

0.02908 

0.03199 

0.03490 

88 

2 

0.03490 

0.03781 

0.04071 

0.04362 

0.04653 

0.04943 

0.05234 

87 

3 

0.05234 

0.05524 

0.05814 

0.06105 

0.06395 

0.06685 

0.06976 

86 

4 

0.06976 

0.07266 

0.07556 

0.07846 

0.08136 

0.08426 

0.08716 

85° 

S'' 

0.08716 

0. 09005 

0.0929s 

0.09585 

0.09874 

0. 10164 

0.104S3 

84 

6 

0. 10453 

0. 10742 

0. 1 1 03 1 

0. 11320 

0. 1 1 609 

0.11898 

Q. I2187 

83 

7 

0. 12187 

0. 12476 

0. 12764 

0.13053 

0.13341 

0.13629 

O.13917 

82 

8 

0.13917 

0. 14205 

0.14493 

0. 14781 

0. 15069 

0.15356 

0.15643 

81 

9 

0.15643 

0.15931 

0. 16218 

0.16505 

0. 16792 

0. 17078 

0.17365 

80° 

10° 

0.17365 

0. 17651 

0.17937 

0. 18224 

0. 18509 

0.18795 

0.19081 

79 

II 

0. I 908 I 

0. 19366 

0.19652 

0.19937 

0. 20222 

0.20507 

0.20791 

78 

12 

0. 20791 

0. 21076 

0.21360 

0.21644 

0.21928 

0.22212 

0.2249s 

77 

13 

0.2249s 

0.22778 

0.23062 

0.23345 

0.23627 

0. 23910 

0. 24192 

76 

14 

0.24192 

0.24474 

o.;?4756 

0.25038 

0.25320 

0.25601 

0.25882 

75° 

15° 

0.25882 

0.26163 

0.26443 

0. 26724 

0. 27004 

0.27284 

0.27564 

74 

i6 

0.27564 

0.27843 

0.28123 

0. 2S402 

0.286S0 

0.28959 

0.29237 

73 

17 

0.29237 

0.29515 

0.29793 

0.30071 

0.30348 

0.30625 

0.30902 

72 

i8 

0.30902 

0.31178 

0.31454 

0.31730 

0.32006 

0.32282 

0.32557 

71 

19 

0-32557 

0.32832 

0.33106 

0.33381 

0.3365s 

0.33929 

0.34202 

70° 

20° 

0.34202 

0.34475 

0.34748 

0.35041 

0.35293 

0.35565 

0.35837 

69 

21 

0.35837 

0.36108 

0.36379 

0. 36650 

0.36921 

0.37191 

0.37461 

68 

22 

0.37461 

0.37730 

0.37999 

0.38268 

0.38537 

0.38805 

0.39073 

67 

23 

0.39073 

0.39341 

0.39608 

0.39875 

0.40142 

0.40408 

0.40674 

66 

24 

0.40674 

0.40939 

0.41204 

0.41469 

0.41734 

0.4199S 

0.42262 

65° 

25° 

0.42262 

0.42525 

0.42788 

0.43051 

0.43313 

0. 43575 

0.43837 

64 

26 

0-43837 

0.44098 

0.44359 

0.44620 

0.44880 

0.45140 

0.45399 

63 

27 

0.45399 

0.45658 

0.45917 

0.46175 

0.46433 

0.46690 

0.46947 

62 

28 

0.46947 

0.47204 

0.47460 

0.47716 

0.47971 

0.48226 

0.48481 

61 

29 

0.48481 

0.4873s 

0.48989 

0.49242 

0.49495 

0.49748 

0. 50000 

60° 

30° 

0.50000 

0.50252 

0.50503 

0.50754 

0.51004 

0.51254 

0.51504 

59 

31 

0.51504 

0.51753 

0. 52002 

0.52250 

0.52498 

0.52745 

0.52992 

58 

32 

0.52992 

0.53238 

0.53484 

0.53730 

0.53975 

0. 54220 

0.54464 

57 

33 

0.54464 

0.54708 

0.54951 

0.55194 

0.55436 

0.55678 

0.55919 

56 

34 

0.55919 

0.56160 

0.56401 

0. 56641 

0.56880 

0.57119 

0.57358 

55° 

35° 

0.57358 

0.57596 

0.57833 

0. 58070 

0.58307 

0.58543 

0.58779 

54 

36 

0.58779 

0. 59014 

0. 59248 

0.59482 

0.59716 

0.59949 

0. 60182 

.53 

37 

0. 60182 

0. 60414 

0. 60645 

0.60876 

0. 61 107 

0.61337 

0. 61566 

52 

38 

0. 61 566 

0.6179s 

0. 62024 

0.62251 

0.62479 

0.62706 

0.62932 

51 

39 

0.62932 

0.63158 

0.633S3 

0.63608 

0.63832 

0.64056 

0.64279 

50° 

40° 

0.64279 

0.64501 

0.64723 

0.64945 

0.65166 

0.65386 

0. 65606 

49 

41 

0.65606 

0.65825 

0.66044 

0.66262 

0.66480 

0.66697 

0. 66913 

48 

42 

0.66913 

0.67129 

0.67344 

0.67559 

0.67773 

0.67987 

0. 68200 

47 

43 

0. 6S200 

0.6S412 

0. 686 24 

0.6S835 

0.69046 

0.69256 

0. 69466 

46 

44 

0.69466 

0.6967s 

0.69883 

0. 70091 

0. 70298 

0.70505 

0. 70711 

45 

60'    50'     40'    30'     20'    10'     0'  A 

.ngle 

COSIMB 


and  Cosines 


Trigonometric  Functions 


^INE 


39 


Angle 

0'     10'      20'     30'     40'      50'     60' 

44 

45" 

0.70711 

0.70916 

0.71121 

0.71325 

0.71529 

0.71732 

0.71934 

46 

0.71934 

0.72136 

0.72337 

0.72537 

0.72737 

0.72937 

0.73135 

43 

47 

0-73135 

0.73333 

0.73531 

0.73728 

0.73924 

0. 74120 

0.74314 

42 

48 

0.74314 

0.74509 

0.74703 

0. 74896 

0. 750S8 

0. 75280 

0.75471 

41 

49 

0-75471 

0.75661 

0.75851 

0. 76041 

0. 76229 

0.76417 

0.76604 

40° 

50° 

0. 76604 

0.76791 

0.76977 

0. 77162 

0.77347 

0.77531 

0.77715 

39 

51 

0.77715 

0.77897 

0. 78079 

0. 78261 

0. 78442 

0. 78622 

0. 78801 

38 

52 

0.78801 

0.78980 

0.79158 

0.79335 

0.79512 

0.79688 

0. 79864 

37 

S3 

0. 79864 

0.80038 

0.80212 

0.80386 

0.80558 

0. 80730 

0.80902 

36 

54 

0.80902 

0.81072 

0.81242 

0.81412 

0.81580 

0.81748 

0. 81915 

35° 

55° 

0.81915 

0.82082 

0.82248 

0.82413 

0.82577 

0.82741 

0.82904 

34 

56 

0.82904 

0. 83066 

0.83228 

0.833S9 

0.83549 

0.83708 

0.83867 

33 

57 

0.83867 

0.84025 

0.84182 

0.84339 

0.84495 

0.84650 

0.84805 

32 

58 

0.84805 

0.84959 

0.85112 

0.85264 

0.85416 

0.85567 

0.85717 

31 

59 

0.85717 

0.85866 

0.S6015 

0.86163 

0.S6310 

0.86457 

0.86603 

30° 

60° 

0.86603 

0.86748 

0.86892 

0.87036 

0.87178 

0.87321 

0.87462 

29 

61 

0.87462 

0.87603 

0.87743 

0.87882 

0. SS020 

0.88158 

0.88295 

28 

62 

0.88295 

0.8S431 

0.88566 

0.88701 

0.88S35 

0.8896S 

o.Sgioi 

27 

63 

0.89101 

0.89232 

0.89363 

0.89493 

0.S9623 

0.89752 

0.89879 

26 

64 

0.89879 

0.90007 

0-90133 

0.90259 

0.90383 

0.90507 

0.90631 

25° 

65° 

0.90631 

0.90753 

0.90875 

0. 90996 

0. 91116 

0.91236 

0.91355 

24 

66 

0-91355 

0.91472 

0.91590 

0. 91706 

0.91822 

0.91936 

0.92050 

23 

67 

0.92050 

0. 92164 

0.92276 

0.92388 

0.92499 

0. 92609 

0.92718 

22 

68 

0.92718 

0.92827 

0.92935 

0.93042 

0.9314S 

0.93253 

0.93358 

21 

69 

0.93358 

0.93462 

0.93565 

0.93667 

0.93769 

0.93869 

0.93969 

20° 

70° 

0.93969 

0.94068 

0.94167 

0.94264 

0.94361 

0. 94457 

0.94552 

19 

71 

0.94552 

0.94646 

0.94740 

0.94832 

0.94924 

0.95015 

0. 95106 

18 

72 

0.95106 

0.95195 

0.95284 

0.95372 

0.95459 

0.95545 

0.95630 

17 

73 

0.95630 

0.95715 

0.95799 

0.95882 

0.95964 

0. 96046 

0.96126 

16 

74 

0.96126 

0.96206 

0.96285 

0.96363 

0.96440 

0.96517 

0.96593 

15° 

75° 

0.96593 

0.96667 

0.96742 

0.96815 

0.96887 

0.96959 

0.97030 

14 

76 

0.97030 

0. 97100 

0. 97169 

0.97237 

0.97304 

0.97371 

0.97437 

13 

77 

0-97437 

0.97502 

0.97566 

0.97630 

0.97692 

0.97754 

0.97815 

12 

78 

0.97815 

0.97875 

0.97934 

0.97992 

0.98050 

0.98107 

0.98163 

II 

79 

0.98163 

0.98218 

0.98272 

0.98325 

0.98378 

0.98430 

0.98481 

10° 

80° 

0.98481 

0.98531 

0.98580 

0.98629 

0.98676 

0.98723 

0.98769 

9 

81 

0.98769 

0.98814 

0.98858 

0.98902 

0.98944 

0.98986 

0.99027 

8 

82 

0. 99027 

0.99067 

0. 99106 

0.99144 

0. 99182 

0.99219 

0.99255 

7 

83 

0.99255 

0. 99290 

0.99324 

0.99357 

0.99390 

0. 99421 

0.99452 

6 

84 

0.99452 

0. 99482 

0.99511 

0.99540 

0.99567 

0.99594 

0.99619 

S° 

85° 

0. 99619 

0.99644 

0.99668 

0.99692 

0.99714 

0.99736 

0.99756 

4 

86 

0.99756 

0.99776 

0.99795 

0.99813 

0.99831 

0.99847 

0.99863 

3 

87 

0.99863 

0.99878 

0.99892 

0.99905 

0.99917 

0.99929 

0.99939 

2 

88 

0.99939 

0.99949 

0.999 58 

0. 99966 

0.99973 

0.99979 

0.99985 

I 

89 

0.99985 

0.99989 

0.99993 

0.99996 

0.99998 

I . 00000 

I . 00000 

0° 

60'     50'    40'     30'     20'     10'     0'  A 

ingle 

COS£N£ 


40 


Trigonometric  Functions 

26. 


Natural  Tangents 


(  -. 

TANGENT 

Angle   o'      lo'     20'     30'     40'     50'     60' 

89 

0° 

0.00000 

0.00291 

0.00582I0. 00873 

0. 01164 

0.0145s 

0.01746 

I 

0.01746 

0.02036 

0.02328I 

0.02619 

0.02910 

0.03201 

0.03492 

88 

3 

0.03492 

0.03783 

0.04075 

0. 04366 

0.04658 

0.04949 

0.05241 

87 

3 

0.05241 

0.05533 

0.05824 

0.061 16 

0.06408 

0.06700 

0.06993 

86 

4 

0.06993 

0.07285 

0.07578 

0.07870 

0.08163 

0.08456 

0.08749 

85° 

5° 

0.08749 

0.09042 

0.09335 

0.09629 

0.09923 

0. 10216 

0. 10510 

84 

6 

0. 10510 

0. 10805 

0. 11099 

0.11394 

0.11688 

0.1 1983 

0.12278 

83 

7 

0. 12278 

0.12574 

0. 12869 

0.13165 

0.13461 

0.13758 

0.14054 

82 

8 

0. 14054 

0.1435 1 

0. 14648 

0.14945 

0.15243 

0.15540 

0.15838 

81 

9 

0.15838 

0.16137 

0.16435 

0.16734 

0.17033 

0.17333 

0.17633 

80° 

10° 

0.17633 

0.17933 

0.18233 

0.18534 

0.18835 

0.19136 

0.19438 

79 

II 

0. 19438 

0. 19740 

0. 20042 

0.20345 

0.20648 

0.20952 

0.21256 

78 

12 

0. 2125 

0.21560 

0. 21864 

0. 22169 

0.22475 

0.22781 

0.23087 

77 

13 

0.23087 

0.23393 

0.23700 

0. 24008 

0.24316 

0.24624 

0.24933 

76 

14 

0.24933 

0.25242 

0.25552 

0. 25862 

0.26172 

0.26483 

0.26795 

75° 

15° 

0.26795 

0.27107 

0.27419 

0.27732 

0. 28046 

0.28360 

0.28675 

74 

16 

0.28675 

0. 28990 

0.29305 

0. 29621 

0.29938 

0.3025s 

0.30573 

73 

17 

0.30573 

0.30S91 

0.31210 

0.31530 

0.31850 

0.32171 

0.32492 

72 

18 

0.32492 

0.32814 

0.33136 

0.33460 

0.33783 

0. 34108 

0.34433 

7X 

19 

0.34433 

0.34758 

0.35085 

0.35412 

0.35740 

0.36068 

0.36397 

70° 

20° 

0.36397 

0.36727 

0.37057 

0.373S8 

0.37720 

0.38053 

0.38386 

69 

21 

0.38386 

0.38721 

0.3905s 

0.39391 

0.39727 

0. 40065 

0.40403 

68 

22 

0.40403 

0.40741 

0. 41081 

0.41421 

0.41763 

0.42105 

0.42447 

67 

23 

0.42447 

0.42791 

0.43136 

0.43481 

0.43828 

0.44175 

0.44523 

66 

24 

0.44523 

0.44872 

0.45222 

0.45573 

0.45924 

0.46277 

0.46631 

65° 

25° 

0. 46631 

0.46985 

0.47341 

0.47698 

0.48055 

0.48414 

0.48773 

64 

26 

0.48773 

0.49134 

0.49495 

0.49858 

0.50222 

0.50587 

0.50953 

63 

27 

0.50953 

0.51320 

0.51688 

0.52057 

0.52427 

0.52798 

0.53171 

62 

28 

0.53171 

0-53545 

0.53920 

0. 54296 

0.54673 

0.55051 

0.55431 

61 

29 

0.55431 

0.55812 

0.56194 

0.56577 

0. 56962 

0.57348 

0.57735 

60° 

30° 

0.5773s 

0.58124 

0.58513 

0.58905 

0.59297 

0. 59691 

0.60086 

59 

31 

0. 600S6 

0.60483 

0.60881 

0.612S0 

0.61681 

0.62083 

0.62487 

58 

32 

0.62487 

0.62892 

0.63299 

0.63707 

0.64117 

0.64528 

0.64941 

57 

33 

0. 64941 

0.65355 

0.65771 

0. 661 89 

0.66608 

0.67028 

0.67451 

56 

34 

0.67451 

0.67875 

0.68301 

0.68728 

0.69157 

0.69588 

0.70021 

55° 

35° 

0. 70021 

0.70455 

0. 70891 

0.71329 

0.71769 

0. 72211 

0.72654 

54 

36 

0.72654 

0.73100 

0.73547 

0.73996 

0.74447 

0. 74900 

0.7535s. 

53 

37 

0.7535s 

0.75812 

0.76272 

0.76733 

0.77196 

0. 77661 

0. 78129 

52' 

38 

0.78129 

0.78598 

0. 79070 

0.79544 

0.80020 

0.80498 

0.80978 

51 

39 

0.80978 

0.81461 

0.81946 

0.82434 

0.82923 

0.83415 

0.83910 

50° 

40° 

0.83910 

0.84407 

0.  ■?49o6 

0.85408 

0.85912 

0.86419 

0.86929 

49 

41 

0.86929 

0.87441 

0.87955 

0.88473 

0.88992 

0.89515 

0.90040 

48 

42 

0.90040 

0.90569 

0.91099 

0.91633 

0.92170 

0.92709 

0.93252 

47 

43 

0.93252 

0.93797 

0.94345 

0. 94896 

0.95451 

0.96008 

0.96569 

46 

44 

0. 96569 

0.97133 

0.97700 

0.98270 

0.98843 

0.99420 

1 . 00000 

45° 

ingle 

60'     50'      40'     30'      20'      10'      0'   / 

COTANG£NT 


Trigonometric  Fdnctions 


41 


and  Cotangents 


TANGENT 

Angle 

;   0'     10'     20'    30'     40'     50'    60' 

45"! 

I. 00000' 

1.00583 

1.01170 

1.01761 

1-02355 

1 .02952 

1-03553 

44 

46   11.03553 

I. 04158 

I .04766 

1.05378 

1.05994 

1 .06613 

1.07237 

43 

47   1.07237 

1.07S64 

I .0S496 , 

1.09131, 

1.09770 

1 . 10414 

1 . 11061 

42 

48 

I. iio6i 

1.11713 

I. 12369 

1.13029 

1.13694 

1-14363 

1-15037 

41 

49 

1.15037 

I-15715 

I. 16398 

I. 17085 

1.17777 

1.18474 

1-19175 

40° 

50°  I. "^175 

I. 19882 

1.20593 

1 . 21310 

1.22031 

1.22758 

1-23490 

39 

51  4.23490 

1.24227 

I . 24969 

1.25717 

1.26471 

1.27230 

1-27994 

38 

52   1.27994 

1.28764 

I. 29541 

1-30323 

1.31110 

1.31904 

1-32704 

37 

53   1.32704 

I. 3351 1 

1.34323 

I-35142 

1.35968 

1 . 36800 

1-37638 

36 

54 

1.37638 

1.38484 

1.39336 

I. 40195 

I .41061 

1.41934 

I. 42815 

35° 

55°ii.428is 

1.43703 

1.44598 

I. 45501 

I . 46411 

1-47330 

1.48256 

34 

56 

1.48256 

1.49190 

I. 50133 

1.510S4 

1.52043 

1.53010 

1.53987 

33 

57 

1.53987 

1.54972 

1.55966 

1.56969 

1-57981 

I . 59002 

1 . 60033 

32 

58 

I . 60033 

I. 61074 

I. 62125 

I. 63185 

1.64256 

1-65337 

1.66428 

31 

59 

1.66428 

1.67530 

1.68643 

1 . 69766 

I. 70901 

1.72047 

1.73205 

30° 

60° 

1.73205 

1.74375 

1.75556 

r. 76749 

1.77955 

I. 79174 

I . 80405 

29 

61 

1.80405 

I . 81649 

1.82906 

I. 84177 

1.85462 

1.86760 

1.88073 

28 

62 

1.8S073 

I . 89400 

I. 90741 

1.9209S 

1.93470 

1.94858 

1.96261 

27 

63 

I .96261 

1.97680 

I .99116 

2.00569 

2. 02039 

2.03526 

2.05030 

26 

64 

2.05030 

2.06553 

2.08094 

2.09654 

2. 11233 

2. 12832 

2.14451 

25° 

65° 

2.14451 

2. 16090 

2.17749 

2.19430 

2.21132 

2.22857 

2. 24604 

24 

66 

2.24604 

2.26374 

2. 28167 

2 . 29984 

2.31826 

2.33693 

2.35585 

23 

67 
68 

2.35585 
2.47509 

2.37504 
2.49597_ 

2.39449 
2.5171S 

2.41421 
2.53865 

2.43422 

2.45451 

2.47509 
2.60509 

22 
21 

2. 56046 

2.58261 

69 

2.60509 

2.62791 

2.65109 

2.67462 

2.69853 

2.72281 

2.74748 

20° 

70° 

2.747.4'  '2.77254 

2.79802 

2.82391 

2.85023 

2.87700 

2.90421 

19 

71 

2.90^:-   J.  93189 

2.96004 

2.98869 

3-01783 

3.04749 

3.0776S 

18 

72 

3-0776' 

;i .  10842 

3-13972 

3-17159 

3.20406 

3-23714 

3-27085 

17 

73 

3-2708; 

3.30521 

3-34023 

3-37594 

3-41236 

3-44951 

3-48741 

16 

74 

3-48741 

3.52609 

3-56557 

3.60588 

3.64705 

3.68909 

3-73205 

15° 

75° 

3-73205 

3-77595 

3-82083 

3.86671 

3-91364 

3-96165 

4.01078 

14 

76 

4.01078 

4.06107 

4. 11256 

4.16530 

4-21933 

4.27471 

4.33148 

13 

77 

4.33148 

4.38969 

4.44942 

4.51071 

4-57363 

4.63825 

4.70463 

12 

78 

4.70463 

4.77286 

4.84300 

4.91516 

4.98940 

5.06584 

5-14455 

II 

79 

5-14455 

5.22566 

5-30928 

5-39552 

5-48451 

S-57638 

5.67128 

10° 

80° 

5.67128 

5. 76937 

5.87080 

5-97576 

6.08444 

6.19703 

6.31375 

9 

81 

6.31375 

6.43484 

6.56055 

6. 69116 

6.82694 

6.96823 

7.11537 

8 

82 

7. "537 

7.26873 

7.42871 

7-59575 

7-77035 

7-95302 

8.14435 

7 

83 
84 

8.14435 
9-51436 

8.34496 
9.78817 

8.55555 
10.0780 

8.77689 
10.3854 

9.00983 
10.7119 

9-25530 

9-51436 

6 

5° 

11.0594 

11.4301 

8S° 

II. 4301 

11.8262 

12.2505 

12.7062 

13.1969 

13.7267 

14.3007 

4 

86 

14.3007 

14.9244 

15.6048 

16.3499 

17.1693 

18.0750 

19.0811 

3 

87 

19.0811 

20. 2056 

21.4704 

22.9038 

24.5418 

26.4316 

28.6363 

2 

88 

28.6363  31.2416 

34.3678 

38.1885 

42,9641 

49.1039 

57.2900 

I 

69 

57.2900  OS. 7501 

85.9398 

,114.589 

171.885 

343- 774  1   ~ 

0° 

60'     50'     40'      30'     20'      10'      0'  ; 

^ngle 

COTANGENT 


42 


Trigonometric  Functions 
27.  Natural  Trigonometric  Functions 


Angle 

Arc 

Sin 

Tan 

Sec 

Cosec 

Cot 

Cos 

1° 

0.017s 

0.017s 

0.0175 

I . ooos 

57.299 

57.290 

0.9998 

1-5533 

89 

2   0.0349 

0.0349 

0.0349 

I .0006 

28. 654 

28.636 

0.9994 

1-5359 

88 

3 

1 
0.0524 

0.0523 

0.0524 

I . 0014 

19.107 

19. 081 

0. 99S6 

1.5184 

87 

4 

0.0698 

0.0698 

0.0699 

I .0024 

14.336 

14.301 

0.9976 

1.5010 

86 

5 

0.0873 

0.0872 

0.0875 

1.0038 

11.474 

11.430 

0. 9962 

1-4835 

8s° 

6° 

0. 1047 

0.1045 

0. 1051 

1.0055 

9.5668 

9.5144 

0.9945 

I. 4661 

84 

7 

0. 1222 

0. 1219 

0. 1228 

1.0075 

8.205s 

8.1443 

0.9925 

1 . 4486 

83 

8 

0.1396 

0.1392 

0. 1405 

I .0098 

7.1853 

7.1154 

0.9903 

1.4312 

8a 

9 

0.1571 

0. 1564 

0. 1584 

I. 0125 

6.3925 

6.3138 

0.9877 

1.4137 

81 

10 

0.1745 

0.1736 

0.1763 

I. 0154 

5.7588 

5.6713 

0.9848 

1-3963 

80° 

11° 

0. 1920 

0. 1908 

0.1944 

I. 0187 

5. 2408 

5.1446 

0.9816 

1.3788 

79 

12 

0.2094 

0.2079 

0. 2126 

1.0223 

4.8097 

4.7046 

0.9781 

1.3614 

78 

13 

0. 2269 

0. 2250 

0.2309 

1.0263 

4.4454 

4.331s 

0.9744 

1.3439 

77 

14 

0.2443 

0.2419 

0.2493 

1.0306 

4.1336 

4.0108 

0.9703 

1.3265 

76 

15 

0.2618 

0.2588 

0.2679 

1-0353 

3  8637 

3.7321 

0.9659 

1 . 3090 

75° 

16° 

0.2793 

0.2756 

0.2867 

I . 0403 

3.6280 

3.4874 

0.9613 

I. 2915 

74 

17 

0. 2967 

0.2924 

0.3057 

1.0457 

3.4203 

3.2709 

0.9563 

1.2741 

73 

18 

0.3142 

0.3090 

0.3249 

1. 05 1 5 

3.2361 

3.0777 

0.9511 

1.2566 

72 

19 

0.3316 

0.3256 

0.3443 

1.0576 

3.0716 

2.9042 

0.9455 

1.239? 

71 

20 

0.3491 

0.3420 

0.3640 

1.0642 

2.9238 

2.7475 

0.9397 

1. 2217 

70° 

21° 

0.3665 

0.3584 

0.3839 

1.0711 

2.7904 

2.6051 

0.9336 

I . 2043 

69 

22 

0.3840 

0.3746 

0. 4040 

1.0785 

2.669s  2.4751 

0.9272 

1.1868 

68 

23 

0.4014 

0.3907 

0.4245 

1.0864 

2.5593,2.3559 

0.9205 

1.1694 

67 

34 

0.4189 

0.4067 

0.4452 

1.0946 

2.4586 

2. 2460 

0.9135 

1-1519 

66 

25 

0.4363 

0.4226 

0.4663 

I. 1034 

2.3662 

2. 1445 

0.9063 

1-1345 

65° 

26° 

0.4538 

0.4384 

0.4877 

I. 1126 

2.2812 

2.0503 

0.8988 

1. 1 1 70 

64 

27 

0.4712 

0.4540 

0.5095 

I. 1223 

2. 2027 

1.9626 

0. 8910 

1.0996 

63 

28 

0.4887 

0.4695 

0.5317 

1.1326 

2.1301 

1.8807 

0.8829 

1.0821 

6a 

29 

0. 5061 

0.4848 

0.5543 

I. 1434 

2.0627 

I . 8040 

0.8746 

1.0647 

61 

30 

0.5236 

0. 5000 

0.5774 

1.1547 

2 . 0000 

1.7321 

0.8660 

1.0472 

60° 

31° 

0.5411 

0.5150 

0,6009 

I. 1666 

I. 9416 

I . 6643 

0.8572 

1.0297 

59 

32 

0.5585 

0.5299 

0.6249 

1.1792 

I. 8871 

I . 6003 

0.8480 

1:0123 

58 

33 

0. 5760 

0.5446 

0.6494 

1-1924 

I. 8361 

1.5399 

0.8387 

0.9948 

57 

34 

0.5934 

0.5592 

0.6745 

I . 2062 

1.7883 

1.4826 

0. 8290 

0.9774 

56 

35 

0.6109 

0.5736 

0.7002 

1.2208 

1-7434 

I. 4281 

0.8192 

0.9599 

55° 

36° 

0.6283 

O.587S 

0.7265 

I. 2361 

1.7013 

1.3764 

0.8090 

0.9425 

54 

37 

0.6458 

0.6018 

0.7536 

1.2521 

I. 6616 

1.3270 

0.7986 

0.9250 

53 

38 

0.6632 

0.6157 

0.7813 

I. 2690 

1.6243 

I. 2799 

0.7880 

0. 9076 

52 

39 

0.6807 

0.6293 

0.8098 

I . 2868 

I . 5890 

I . 2349 

0.7771 

0.S901 

51 

40 

0.6981 

0.6428 

0.S391 

1.3054 

1-5557 

I. 1918 

0.7660 

0.8727 

50° 

41° 

0.7156 

0.6561 

0.8693 

1.3250 

1-5243 

1.1504 

0.7547 

0.8552 

49 

42 

0.7330 

0.6691 

0. 9004 

1.3456 

1.4945 

1. 1106 

0.7431 

0.8378 

48 

43 

0.7505 

0.6820 

0.9325 

1.3673 

1 . 4663 

1.0724 

0.7314 

0.8203 

47 

44 

0.7679 

0.6947 

0.9657 

1.3902 

I . 4396 

1.0355 

0.7193 

0.8029 

46 

45 

0.7854 

0.7071 

I .0000 

I. 4142 

1-4142 

I . 0000 

0.7071 

0.7854 

45° 

Cos 

Cot 

Cosec 

Sec 

Tan 

Sin 

Arc 

Angle 

Trigonometric  Functions  7"%  43 


28.  Explanations 


Table  25  gives  Natural  Sines  and  Cosines  of  angles  for  every 
10  minutes  from  0°  0'  to  90°  0'.  When  the  sine  is  sought,  the  angle, 
or  argument,  is  to  be  looked  for  at  the  left-hand  side  and  at  the 
top  of  the  page;  when  the  cosine  is  sought,  the  angle  is  to  looked 
for  at  the  right-hand  side  and  at  the  foot  of  the  page.  Thus  the 
sine  of  64°  50'  is  0.90507,  but  the  cosine  of  64°  50'  is  0.42525. 
Again,  the  number  0.36108  is  seen  to  be  the  sine  of  21°  10'  or  the 
cosine  of  68°  50'. 

Table  26,  which  is  arranged  like  table  25,  gives  Natural  Tan- 
gents and  Cotangents  of  angles. 

Interpolation  in  these  tables  can  be  made  for  a  given  angle  like 
13°  27'  as  explained  in  Art.  3,  but  the  last  figure  of  the  function  may 
be  sometimes  one  unit  in  error  for  the  sine  and  cosine,  and  more 
than  one  unit  for  a  tangent  of  an  angle  greater  than  60°  or  for  a 
cotangent  of  an  angle  less  than  30°.  For  example  the  table  gives 
sin  14°  12' =0.24530  and  cot  14°  12' =3.95205,  the  former  being 
in  error  one  unit  in  the  last  place  and  the  latter  nine  units. 

Table  27  gives  all  common  Trigonometric  Functions  to  four 
places.  Here  Arc  is  the  length  of  the  arc  of  the  angle  in  a  circle 
of  radius  unity;  thus  arc  25°  =0.4363,  as  may  be  otherwise  found 
from  Table  22.  The  secant  is  the  reciprocal  of  the  cosine  and  the 
cosecant  of  the  sine.  Interpolation  need  rarely  be  made  in  this 
table.  When  the  angle  is  less  than  45°  look  for  it  at  the  left- 
hand  side  of  the  table  and  for  the  name  of  the  functions  at  the  top; 
for  angles  between  45°  and  90°  look  for  the  angle  at  right-hand 
side  and  for  the  name  of  the  function  at  the  foot.  Thus,  sin  41° 
=  0.6561,  cos  50°  =0.6428,  sec  75°  =3.8637. 

Inverse  Interpolation  is  the  process  of  finding  an  argument 
from  a  given  value  of  a  function.  If  the  sine  be  given  as  0.70916, 
the  corresponding  angle  is  seen  from  Table  25  to  be  45°  10',  and 
here  no  interpolation  is  necessary.  But  let  the  sine  0.70987 
be  given,  then  the  angle  is  seen  to  lie  between  45°  10'  and  45°  20'; 
the  difference  of  the  sines  of  these  angles  is  0.00205,  hence  the 


44  Trigonometric  Functions 

difference  for  1'  is  0.000205;  now  the  given  sine  is  greater  than  the 
sine  of  45°  10'  by  0.00071,  then  71/20.7  =3.5,  so  that  the  required 
angle  is  45°  13. '5.  It  is  important  to  note  whether  or  not  the 
values  of  the  function  increase  with  the  argument;  thus,  if  the 
cosine  0.94698  be  given,  the  angle  is  seen  to  be  less  than  18°  50'  and 
more  than  18°  40',  so  that  the  computed  difference  is  to  be  sub- 
tracted; here  the  angle  will  be  found  to  be  18°  47'  closely. 

29.  Exercises 

1.  Find  the  values  of  the  following  functions  to  five  decimal  places: 

sin  25°  20'=  cos  25°  20'  = 

sin  85°  40'=  cos  85°  40'  = 

sin  77°  34'=  cos  77°  34'  = 

2.  Find  the  angle  whose  sine  is  0.39700.  Also  the  angle  whose 
tangent  is  1.24312. 

3.  Find  the  values  of  the  following  functions  to  four  decimal  places: 

sin  30°  =  cos  30°  =  tan  30°  = 

sin  60°  =  cos  60°  =  tan  60°  = 

sec  30°  =  cosec  60°  =  arc  60°  = 

4.  Multiply  the  tangent  of  11°  20'  by  the  cotangent  of  the 
same  angle. 

5.  Find  sin  45°  and  cos  45°  by  Table  27,  and  then  multiply  them 
together. 

6.  Find  the  sine  and  cosine  of  17°,  square  each  by  help  of  Table  7, 
and  then  add  these  squares. 

7.  Find  the  value  of  arc  78°  by  Table  22  and  also  by  Table  27. 

8.  Find  the  values  of  the  following  functions  to  five  decimal  places: 

cos  32°  33'=  cot  32°  33'  = 

sin  57°  27'=  tan  57°  27'  = 

cot  40°  15'=  cot  49°  45'  = 

0.  Test  the  equation  cos-  0  — sin-  9  =  cos  20  by  assuming'a  value  of 
e,  taking  the  functions  from  Table  27,  and  the  squares  from  Table  7. 

10.  A  vertical  post  3.64  foot  high  casts  a  shadow  10.0  feet  long 
on  level  ground.     IIow  high  is  the  sun  above  the  hori/on? 

11.  Find  thefangles  whose  sines  ^re  0.5000,  0.8660,  and  0.9979; 
also  the  angles  whose  tangents  are  0.1,  0.3,  0.5,  0.7,  and  0.9;  also 
the  angles  whose  tangents  are  1.0,  2.0,  3.0,  and  4.0. 


Chapter  5 
LOGARITHMIC  TABLES 


46 


Logarithmic  Tables 

30.  Common  Logarithms 


« 

lO 

0123456789 

00000 

00432 

00860 

01284 

01703 

02 1 1^ 

02531 

02938 

03342 

03743 

II 

04139 

04532 

04922 

05308 

05690 

06070 

^446 

06819 

07188 

07555 

12 

07918 

08279 

08636 

o§99i 

09342 

09691 

10037 

10380 

10721 

11059 

13 

"394 

11727 

12057 

123S5 

12710 

13033 

13354 

13672 

13988 

14301 

.  14 

14613 

14922 

15229 

15534 

15836 

16137 

16435 

16732 

17026 

17319 

IS 

17609 

17898 

18184 

18469 

18752 

19033 

19312 

19590 

19866 

20140 

i6 

20412 

20683 

209'52 

21219 

214S4 

21748 

22011 

22272 

22531 

22789 

17 

2304s 

23300 

23553 

23805 

2405s 

24304 

24551 

24797 

25042 

25285 

i8 

25527 

25768 

26007 

26245 

26482 

26717 

26951 

271S4 

27416 

27646 

19 

27875 

28103 

28330 

28556 

28780 

29003 

29226 

29447 

29667 

29885 

20 

30103 

30320 

30535 

30750 

30963 

31175 

31387 

31597 

31806 

32015 

21 

32222 

32428 

32634 

3283S 

33041 

33244 

33445 

33646 

33846 

34044 

22 

34242 

34439 

3463  s 

34830 

35025 

35218 

3S4II 

35603 

35793 

359S4 

33 

36173 

36361 

36549 

36736 

36922 

37107 

37291 

37475 

37658 

37840 

24 

38021 

38202 

38382 

38561 

38739 

38917 

39094 

39270 

39445 

39620 

25 

39794 

39967 

40140 

40312 

40483 

40654 

40824 

40993 

41162 

41330 

26 

41497 

41664 

41830 

41996 

42160 

42325 

42488 

42651 

42813 

42975 

27 

43136 

43297 

43457 

43616 

43775 

43933 

44091 

44248 

44404 

44560 

28 

44716 

44871 

45025 

45179 

45332 

45484 

45637 

45788 

45939 

46090 

29 

46240 

46389 

46538 

46687 

46S35 

46982 

47129 

47276 

47422 

47567 

30 

47712 

47857 

48001 

48144 

48287 

48430 

4S572 

48714 

4885s 

48996 

31 

49136 

49276 

49415 

49554 

49693 

49S31 

49969 

50106 

50243 

50379 

32 

50515 

50651 

50786 

50920 

5105s 

51188 

51322 

5145s 

51587 

51720 

33 

51851 

51983 

52114 

52244 

52375 

52504 

52634 

52763 

52892 

53020 

34 

53148 

53275 

53403 

53529 

53656 

53782 

53908 

54033 

54158 

54283 

35 

54407 

54531 

54654 

54777 

54900 

55023 

55145 

55267 

55388 

55509 

36 

55630 

55751 

55871 

55991 

56110 

56229 

56348 

56467 

56585 

56703 

37 

56820 

56937 

57054 

57171 

57287 

57403 

57519 

57634 

57749 

57S64 

38 

57978 

58092 

58206 

58320 

58433 

58546 

58659 

58771 

58883 

58995 

39 

59106 

59218 

59329 

59439 

59550 

59660 

59770 

59879 

5998S 

60097 

40 

60206 

60314 

60423 

60531 

60638 

60746 

60853 

60959 

61066 

61172 

41 

61278 

613S4 

61490 

61595 

61700 

61805 

61909 

62014 

62118 

62221 

42 

62325 

62428 

62531 

62634 

62737 

62839 

62941 

63043 

63144 

63246 

43 

63347 

63448 

63548 

63649 

63749 

63849 

63949 

6404S 

64147 

64246 

44 

64345 

64444 

64542 

64640 

64738 

64836 

64933 

65031 

65128 

65225 

45 

65321 

65418 

65514 

65610 

65706 

65801 

65S96 

65992 

66087 

661S1 

46 

66276 

66370 

66464 

66558 

66652 

66745 

66839 

66932 

67025 

67117 

47 

67210 

67302 

67394 

67486 

67578 

67669 

67761 

67852 

67943 

68034 

48 

68124 

68215 

6S305 

68395 

68485 

68574 

68664 

68753 

68842 

68931 

49 

69020 

69108 

69197 

69285 

693*73 

69461 

69548 

69636 

69723 

69810 

50 

6989^ 

69984 

70070 

70157 

70243 

70329 

70415 

70501 

70586 

70672 

51 

70757 

70842 

70927 

71012 

71096 

7 II 81 

71265 

71349 

71433 

71517 

52 

71600 

71684 

71767 

71850 

71933 

72016 

72099 

72x81 

72263 

72346 

53 

72428 

72509 

72591 

72673 

72754 

72835 

72916 

72997 

73078 

73159 

54 

73239 

73320 

73400 

73480 

73560 

73640 

73719 

73799 

73878 

73957 

0123456789 

Logarithmic  Tables 
of  Numbers  from  000  to  999 


47 


n 

01      2      34     56789J 

55 

74036 

74115 

74194 

74273 

743SI 

74429 

74507 

74586 

74663 

74741 

56 

74819 

74896 

74974 

75051 

75128 

75205 

75282 

75358 

75435 

755" 

57 

75^87 

75664 

7  5  740 

75815 

75891 

75967 

76042 

76118 

76193 

76268 

58 

76343 

7641S 

76492 

76567 

76641 

76716 

76790 

76S64 

76938 

77012 

59 

77085 

77159 

77232 

77305 

77379 

77452 

77525, 

77597 

77670 

77743 

6o 

77815 

77887 

77960 

78032 

78104 

78176 

78247 

78319 

78390 

78462 

6i 

78533 

78604 

78675 

78746 

78817 

78888 

78958 

79029 

79099 

79169 

62 

79239 

79309 

79379 

79449 

79518 

7958S 

79657 

79727 

79796 

79865 

63 

79934 

80003 

S0072 

80140 

80209 

80277 

80346 

80414 

80482 

8oS5'ii 

64 

80618 

S06S6 

80754 

80821 

80889 

80956 

81023 

81090 

81158 

8122J; 

65 

812^1 

81358 

81425 

81491 

S1558 

81624 

81690 

81757 

81823 

81889 

66 

81954 

82020 

82086 

82151 

82217 

82282 

82347 

82413 

82478 

82543 

67 

82607 

82672 

82737 

82802 

82866 

82930 

82995 

83059 

83123 

83187 

68 

83251 

83315 

83378 

83442 

83506 

83569 

83632 

83696 

83759 

S3822 

69 

83885 

83948 

8401 1 

84073 

84136 

84198 

84261 

84323 

84386 

84448 

70 

84510 

84572 

84634 

84696 

84757 

84819 

84880 

84942 

85003 

85065 

71 

85126 

85187 

85248 

85309 

85370 

85431 

85491 

85552 

85612 

S5673 

72 

85733 

85794 

85854 

85914 

85974 

86034 

86094 

86153 

86213 

86273 

73 

86332 

86392 

86451 

86510 

86570 

86629 

86688 

86747 

86806 

86864 

74 

86923 

86982 

87040 

87099 

87157 

87216 

87274 

87332 

87390 

87448 

75 

87506 

87564 

87622 

87679 

87737 

87795 

87852 

87910 

87967 

88024 

76 

88081 

88138 

88195 

88252 

88309 

88366 

88423 

88480 

88536 

88593 

77 

88649 

88705 

88762 

88818 

88874 

88930 

88986 

89042 

89098 

89154 

78 

89209 

89265 

89321 

89376 

89432 

89487 

S9542 

89597 

89653 

89708 

79 

89763 

89818 

89873 

89927 

89982 

90037 

90091 

90146 

90200 

90255 

80 

90309 

90363 

90417 

90472 

90526 

90580 

90634 

90687 

90741 

9079s 

81 

90S49 

90903 

90956 

91009 

91062 

91116 

91169 

91222 

91275 

91328 

82 

91381 

91434 

91487 

91540 

91593 

91645 

91698 

91751 

91803 

91855 

83 

91908 

91960 

92012 

92065 

92117 

92169 

92221 

92273 

92324 

92376 

84 

92428 

92480 

92531 

92583 

92634 

92686 

92737 

92788 

92840 

92891 

85 

92942 

92993 

93044 

93095 

93146 

93197 

93247 

93298 

93349 

93399 

86 

93450 

93500 

93551 

93601 

93651 

93702 

93752 

93802 

93852 

93902 

87 

93952 

94002 

94052 

94101 

941,51 

94201 

94250 

94300 

94349 

94399 

88 

9444S 

94498 

94547 

94596 

94645 

94694 

94743 

94792 

94841 

94S90 

89 

94939 

94988 

95036 

95085 

95134 

95182 

95231 

95279 

95328 

95376 

90 

95424 

95472 

95521 

95569 

95617 

95665 

95713 

95761 

95809 

95856 

91 

95904 

95952 

95999 

96047 

9609s 

96142 

96190 

96237 

96284 

96332 

92 

96379 

96426 

96473 

96520 

96567 

96614 

96661 

96708 

96755 

96S02 

93 

96848 

96895 

96942 

96988 

97035 

97081 

97128 

97174 

97220 

97267 

94 

97313 

97359 

97405 

97451 

97497 

97543 

97589 

97635 

97681 

97727 

95 

97772 

97818 

97864 

97909 

97955 

98000 

98046 

98091 

98137 

98182 

96 

98227 

98272 

98318 

98363 

98408 

98453 

98498 

98543 

98588 

98632 

97 

98677 

98722 

98767 

98811 

98856 

98900 

98945 

98989 

99034 

99078 

98 

99123 

99167 

99211 

99255 

99300 

99344 

99388 

99432 

99476 

99520 

99 

99564 

99607 

99651 

99695 

99739 

99782 

99826 

99870 

99913 

99957 

01     234     56     f           89 

J 

48 


Logarithmic  Tables 

31.  Common  Logarithms 

tOG   SINE 


Angle   o'     lo'     20' 

30'     40' 

50' 

60' 

0° 

.  —  00 

3-46373 

3-76475 

3.94084 

2.06578 

2.16268 

2. 24186 

89 

I 

2.24186 

2.30879 

2.36678 

2.41792 

2 .46366 

2.50504 

2.54282 

88 

2 
3 

2.54282 
2.71880 

2-57757 
2.74226 

5.60973 
2.76451 

2.63968 

2.78568 

2 . 66769 

2.80585 

2 . 69400 
2.82513 

2.71880 

87 

2.84358 

86 

4 

0 

2.84358 

2.86128 

2.87829 

2.89464  »2.9i040 

2.92561 

2.94030 

85° 

84 
83 

S 
6 

7 

2.94030 
T. 01923 

T.0S589 

2.95450 
I. 03109 
1.09606 

2.96825 
T. 04262 
T. 10599 

2.98157  - 

2  •  994J0 

1.00704 
T. 07548 

I. 01923 

1.08589 

1.05386 
1.11570 

1. 0648 1 
1. 12519 

I. 13447 

I- 14356 

82 

8 
9 

I. 14356 
1 .19433: 

1-15245 
1.20223 

I . 16116 
T. 20999 

I . 16970 
I . 21761 

T. 17807 
1.22509 

I. 18628 

I -19433 

81 
80° 

1.23244 

1.23967 

10° 

II 

12 

T. 23967 
1.28060 
T. 31788 

T. 24677 
1.28705 
1-32378 

1-25376 
1.29340 

T. 26063 
I . 29966 

X 26739 

T. 27405 

T. 28060 

79 
78 
77 

1.30582 
'T.  34100 

I. 31189 
T- 34658 

I. 31788 
r-35209 

1.32960 

1-33534 

13 

1.35209 

1-35752 

1.36289 

T. 36819 

I. 37341 

1-37858 

1.38368 

76 

14 

T. 38368 

T. 38871 

1.39369 

I.39S60 

1.40346 

1.40825 

1.41300 

75° 

15° 

T. 41300 

T. 41768 

1.42232 

I . 42690 

1-43143 

1-43591 

1.44034 

74 

16 

1.44034 

1.44472 

1.44905 

1-45334 

1-45758 

1.4617S 

1.46594 

73 

17 

T. 46594 

1.47005 

1.47411 

I. 47814 

I. 48213 

T. 48607 

1 .48998 

72 

18 

1.48998 

1.49385 

1.49768 

I . 50148 

1-50523 

T. 50896 

T. 51264 

71 

19 

T. 51264 

I. 51629 

T.51991 

1-52350 

1-52705 

1-53056 

1-53405 

70° 

20° 

1-53405 

1.53751 

1-54093 

T- 54433 

T- 54769 

T. 55102 

1-55433 

69 

21 

1-55433 

1-55761 

1.56085 

1.56408 

1.56727 

1-57044 

1-57358 

68 

22 

1-57358 

1.57669 

1-57978 

1.582S4 

T. 58588 

T. 58889 

I . 591S8 

67 

23 

T. 59188 

1.59484 

1-59778 

I . 60070 

1.60359 

I . 60646 

T. 60931 

66 

24 

I. 6093 I 

1.61214 

I. 61494 

T. 61773 

1.62049 

1.62323 

1-62595 

65° 

25° 

1-62595 

1.62865 

I -63133 

T. 63398 

T. 63662 

T. 63924 

T. 64184 

64 

26 

I. 64184 

1.64442 

T. 64698 

1-64953 

1.65205 

1.65456 

1-65705 

63 

37 

1-65705 

1-65952 

I. 66197 

I . 66441 

1.66682 

I .66922 

1.67161 

62 

28 

1.67161 

T.6739S 

1-67633 

1.67866 

1.68098 

T. 68328 

1-68557 

61 

29 

1-68557 

1.68784 

I . 69010 

T. 69234 

1.69456 

1.69677 

1.69897 

60° 

30° 

T. 69897 

1.70115 

1-70332 

1-70547 

I. 70761 

T. 70973 

1.71184 

59 

31 

T. 71184 

1-71393 

I. 71602 

I . 71809 

T. 72014 

T. 72218 

1.72421 

58 

32 

T. 72421 

1.72622 

1.72823 

T. 73022 

T. 73219 

T. 73416 

T. 73611 

57 

33 

1.73611 

1-73805 

1-73997 

I. 74189 

1.74379 

1.74568 

1.74756 

56 

34 

1-74756 

I  -  74943 

I. 75128 

1-75313 

1.75496 

1.75678 

1.75859 

55° 

35° 

1-75859 

T. 76039 

T. 76218 

1-76395 

T. 76572 

1.76747 

1.76922 

54 

36 

1.76922 

1.77095 

1.77268 

1-77439 

1.77609 

I- 77778 

1-77946 

53 

37 

T. 77946 

I. 78113 

I. 78280 

I ■ 78445 

I. 78609 

1.78772 

1-78934 

52 

38 

1-78934 

T. 79095 

T. 79256 

1-79415 

1.79573 

1-79731 

1.79887 

■51 

39 

1.79887 

I . 80043 

T. 80197 

1-80351 

1.80504 

T. 80656 

T. 80807 

50° 

40° 

1.80807 

1.80957 

T. 81106 

T. 81254 

T. 81402 

T. 81549 

I . 81694 

49 

41 

T. 81694 

T. 81839 

I. 81983 

I . 82126 

T. 82269 

T. 82410 

1-82551 

48 

42 

1-82551 

T. 82691 

1.82830 

T.8296S 

T. 83106 

T. 83242 

1-83378 

47 

43 

1-83378 

I -83513 

T. 83648 

1.83781 

T. 83914 

1 . 84046 

I. 84177 

46 

44 

I. 84177 

1.84308 

1.84437 

T. 84566 

T. 84694 

1.84822 

1.84949 

45° 

60'      50'     4°' 

30'     20' 

10' 

0'  1 

\nglc 

I.OG  COSIMi: 


11 


90--»" 


Logarithmic  Tables 


49 


of  Sines  and  Cosines 


LOG  SINE 


Angl( 
45" 

i      0'             10'     20'     30'     40'     50'     60' 

1.84949 

1.85074 

I . 85200 

1-85324 

T. 85448 

1-85571 

1-85693 

44 

46 

1.85693 

I. 85815 

1-^5936 

T. 86056 

T. 86176 

1.86295 

1.86413 

43 

47 

I. 86413 

1.86530 

T. 86647 

1.86763 

T. 86879 

T. 86993 

T.S7107 

42 

48 

f. 87107 

I. 87221 

1-87334 

1.87446 

1-87557 

1.87668 

T. 87778 

41 

49 

1.87778 

1.87887 

1.87996 

I. 88105 

I. 88212 

1.88319 

T. 88425 

40° 

50° 

1.88425 

T. 88531 

1.88636 

T. 88741 

1.88844 

T. 88948 

I. 89050 

39 

51 

1.89050 

I. 89152 

T. 89254 

T- 89354 

1-89455 

1-S9554 

1.89653 

38 

52 

1-89653 

1.89752 

T. 89849 

1.89947 

I . 90043 

1.90139 

1-90235 

37 

53 

1-90235 

1-90330 

1.90424 

I . 9051S 

I . 90611 

1.90704 

1.90796 

36 

54 

1.90796 

T. 90887 

1.90978 

I. 91069 

1.91158 

I. 91248 

1-91336 

35° 

55° 

1-91336 

I. 91425 

T .  9 1 5 1 2 

I -91599 

T. 91686 

1.91772 

1-91857 

34 

56 

1-91857 

I . 91942 

I . 92027 

1 . 921 1 1 

T. 92194 

1.92277 

1-92359 

33 

57 

1-92359 

T. 92441 

I . 92522 

T. 92603 

1.92683 

1.92763 

I. 92842 

32 

58 

1.92842 

T. 92921 

T. 92999 

1.93077 

I-93154 

1.93230 

1-93307 

31 

59 

1-93307 

1.93382 

1-93457 

1-93532 

1.93606 

1.93680 

1-93753 

30° 

60° 

I-937S3 

T. 93826 

T. 93898 

1.93970 

I. 9404 I 

I . 94112 

T. 94182 

29 

61 

I. 94182 

1.94252 

1-9432-1 

1.94390 

T. 94458 

T. 94526 

1-94593 

28 

62 

1-94593 

I. 94660 

1.94727 

1-94793 

T. 94858 

T. 94923 

1 . 94988 

27 

63 

T. 94988 

1-95052 

1.95116 

1-95179 

T. 95242 

1-95304 

T. 95366 

26 

64 

1-95366 

1-95427 

T.954S8 

1-95549 

1.95609 

T. 95668 

1-95728 

25° 

65° 

1.95728 

T. 95786 

1-95844 

T. 95902 

T. 95960 

T. 96017 

1.96073 

24 

66 

1.96073 

I . 96129 

I. 96 1 85 

T. 96240 

T. 96294 

1.96349 

I . 96403 

23 

67 

I . 96403 

T. 96456 

T. 96509 

T. 96562 

T. 96614 

I . 96665 

1,96717 

22 

68 

T. 96717 

1.96767 

T. 96818 

T. 96868 

T. 96917 

1.96966 

1.97015 

21 

69 

I-97015 

1.97063 

1.97111 

1-97159 

1.97206 

T. 97252 

1.97299 

20  ' 

70° 

1,97299 

1-97344 

1.97390 

1-97435 

T. 97479 

1-97523 

1-97567 

19 

71 

1-97567 

I. 97610 

1-97653 

I . 97696 

1-97738 

1.97779 

1.97S21 

18 

72 

I. 97821 

1.97861 

1.97902 

T. 97942 

1.97982 

1.98021 

1.98060 

17 

73 

1 . 98060 

1.98098 

I. 98136 

I. 98174 

1. 982 1 1 

T. 98248 

T. 98284 

16 

74 

1.982S4 

T. 98320 

1-98356 

I -98391 

1.98426 

1.98460 

1.98494 

15° 

75° 

T. 98494 

1.98528 

I. 98561 

1-98594 

1.98627 

T. 98659 

T. 98690 

14 

76 

T. 98690 

I. 98722 

1-98753 

T. 98783 

I. 98813 

1.98843 

T. 98872 

13 

77 

T.98S72 

T. 98901 

1.98930 

T. 98958 

T. 98986 

I. 99013 

T. 99040 

12 

78 

T. 99040 

1 .  99067 

1.99093 

T.99119 

1-99145 

1.99170 

I-99195 

II 

79 

I-99195 

I. 99219 

1-99243 

1.99267 

1.99290 

1-99313 

1-99335 

10° 

80° 

1-99335 

1-99357 

T-99379 

1.99400 

T. 99421 

T. 99442 

T. 99462 

9 

81 

1.99462 

1.99482 

1.99501 

1.99520 

1-99539 

1-99557 

1-99575 

8 

82 

1-99575 

1-99593 

I .99610 

1.99627 

1.99643 

1.99659 

1.99675 

7 

83 

1-99675 

1.99690 

1-99705 

1.99720 

1-99734 

1.99748^ 

1.99761 

6 

84 

I. 99761 

T-99775 

1.99787 

X . 99800 

1.99812 

1.99823 

1.99834 

5° 

85° 

1.99834 

1-99845 

T. 99856 

T. 99866 

T. 99876 

T. 99885 

T. 99894 

4 

86 

1.99894 

1.99903 

T.99911 

I. 99919 

I . 99926 

1-99934 

1.99940 

3 

87 

1.99940 

1.99947 

1-99953 

1-99959 

T. 99964 

1.99969 

1.99974 

2 

88 

T. 99974 

1.99978 

1.99982 

1.99985 

T. 99988 

1.99991 

1-99993 

I 

89 

1-99993 

1-99995 

1.99997 

1-99998, 

1-99999 

0 . 00000 

0.00000 

0° 

60'     50'     40'      30'     20'      10'      0'  / 

"ingle 

r,OG  coscNs: 


50 


Logarithmic  Tables 

32.  Common  Logarithms 

LOG  TANGENT 


fl-     i^ 


Ang 

e   0'      10'     20'     30'     40'     50'     60' 

89 

0° 

—  CO 

3-46373 

3.76476 

3 • 94086 

1 2.06581 

2.16273 

2.24192 

I 

2. 24192 

2.30888 

2.366S9 

2.41807 

2.46385 

2-50527 

2.54308 

88 

2 

2.54308 

2.57788 

2 . 61009 

2 . 64009 

2.66816 

2.69453 

2.71940 

87 

3 

2.71940 

2.74292 

2.76525 

2. 78649 

2 . 80674 

2.82610 

2.84464 

86 

4 

2.84464 

2.86243 

2-87953 

2.89598 

2-91185 

2.92716 

2-94195 

85° 

5° 

2-94195 

2.95627 

2.97013 

2-98358 

2 .99662 

T. 00930 

r. 02162 

84 

6 

I. 02162 

1-03361 

I .04528 

1 .05666 

T. 06775 

1-07858 

T. 08914 

83 

7 

T. 08914 

1.09947 

T. 10956 

1-11943 

I . 12909 

1-13854 

T. 14780 

82 

8 

T. 14780 

I. 15688 

1-16577 

I- 17450 

T. 18306 

1. 19146 

1. 19971 

81 

9 

J0° 

T.19971 

1.207S2 

1.21578 

T. 22361 

I-23130 

1.23887 

T. 24632 

80° 

T. 24632 

1-25365 

T. 26086 

T. 26797 

1-27496 

T. 28186 

1.28865 

79 

II 

T. 28865 

1-29535 

1-30195 

1.30846 

1.31489 

1.32122 

1-32747 

78 

12 

1-32747 

1-33365 

1-33974 

1-34576 

1-35170 

I-3S757 

1-36336 

77 

13 

1-36336 

1.36909 

1-37476 

1-38035 

1-38589 

1-39136 

1-39677 

76 

14 

1-39677 

T. 40212 

1.40742 

1.41266 

1-41784 

1-42297 

1.42805 

75° 

15° 

T. 42805 

T- 43308 

T. 43806 

T. 44299 

1-44787 

1-45271 

1-45750 

74 

i6 

1-45750 

1.46224 

1.46694 

I. 47160 

1.47622 

I . 48080 

1-48534 

73 

17 

1-48534 

T. 48984 

T. 49430 

1.49872 

1-50311 

T- 50746 

I. 51178 

72 

i8 

1.51178 

1.51606 

1.52031 

1-52452 

1.52870 

1-53285 

1-53697 

71 

19 

1-53697 

1.54106 

1-54512 

1-54915 

1-55315 

1-55712 

I. 56107 

70° 

20° 

T. 56107 

T. 56498 

T. 56887 

1-57274 

1-57658 

1-58039 

T. 58418 

69 

21 

T. 58418 

1-58794 

1.59168 

1 • 59540 

1 ■ 59909 

T. 60276 

I. 60641 

68 

22 

1 . 6064 1 

I . 61004 

T. 61364 

1 . 61722 

T. 62079 

1-62433 

1-62785 

67 

23 

1.62785 

1-63135 

T. 63484 

1.63S30 

1-64175 

T. 645 1  7 

T. 64858 

66 

24 

T.6485S 

I-65197 

1-65535 

T. 65870 

1.66204 

1-66537 

T. 66867 

65° 

25° 

T. 66867 

T. 67196 

1.67524 

T. 67850 

T. 68174 

1.68497 

T. 68818 

64 

26 

T. 68818 

1.69138 

1-69457 

1-69774 

T. 70089 

T. 70404 

T. 70717 

63 

27 

1.70717 

I. 71028 

I-71339 

1.71648 

1-71955 

1.72262 

1.72567 

62 

28 

1.72567 

T. 72872 

1-73175 

1-73476 

1-73777 

1-74077 

1-74375 

61 

29 

1-74375 

1.74673 

T. 74969 

1.75264 

1-75558 

1-75852 

1.76144 

60° 

30° 

I. 76144 

1-76435 

T. 76725 

1.77015 

T- 77303 

1-77591 

1-77877 

59 

31 

T. 77877 

T. 78163 

1. 7844S 

1.78732 

1.79015 

1-79297 

1-79579 

58 

32 

1-79579 

T.  79860 

T. 80140 

I . 80419 

T. 80697 

1.80975 

I. 81252 

57 

33 

I . 81252 

T. 81528 

T.S1803 

T. 82078 

T. 82352 

T. 82626 

1.82899 

56 

34 

T. 82899 

T. 83171 

1-83442 

1-83713: 

T. 83984 

T. 84254 

1-S4523 

55° 

35° 

i-f'4S23 

r. 84791 

1-85059 

T- 85327' 

1-85594 

1.85860 

T. 86126 

54 

36 

T. 86126 

1.86392 

T. 86656 

1.S6921 

T. 87185 

T.S7448 

T. 87711 

53 

37 

T.87711 

1. 87974 

1.88236 

T. 88498' 

T. 88759 

T. 89020 

T. 89281 

52 

38 

T. 89281 

1.89541 

T. 89801 

1.90061 

1.90320 

T. 90578 

1-90837 

51 

39 

1.90837 

T. 91095 

1-91353 

T. 91610 

T.9186S 

T. 92125 

1-92381 

50° 

40° 

1.92381 

T. 92638 

T. 92894 

I-93150 

T. 93406 

T. 93661 

T. 93916 

49 

41 

I. 93916 

1.94171 

T. 94426 

I. 9468 1 

1-94935 

1  95190 

1-95444 

48 

42 

I -95444 

1.95698 

1-95952 

1-96205; 

.1-96459 

1.96712 

T. 96966 

47 

43 

I . 96966 

T. 97219 

1.97472 

1-97725 

1. 97978 

T. 98231 

T.984S4 

46 

44 

T. 98484 

1-98737 

T. 98989 

1.99242  1 

I -99495 

1.99747 

0.00000 

45° 

60'     50'      40'     30'      20'      10'     0'   / 

Ingle 

LOG  COTA^GJiNT 


5    Z,     -^       .    ^;. 


Logarithmic  Tables 
of  Tangents  and  Cotangents 

L,OG  TANGENT 


51 


Ang 

e   0'     10'      20'     30'     40'     50'     60' 

45° 

0.00000 

0.00253 

0.00505 

0.00758 

O.OIOI I 

0. 01263 

0. 01516 

44 

46 

0.01516 

0.01769 

0. 02022 

0.02275 

0.02528 

0. 02781 

0.03034 

43 

47 

0.03034 

0.03288 

0.03541 

0.03795 

0.0404S 

0.04302 

0.04556 

42 

48 

0.04556 

0.04810 

0.05065 

0.05319 

0.05574 

0. 05829 

0.06084 

41 

49 

0.06084 

0.06339 

0.06594 

0. 06850 

0.07106 

0.07362 

0.07619 

40° 

50° 

0.07619 

0.07875 

0.08132 

0.08390 

0.08647 

0.08905 

0.09163 

39 

51 

0.09163 

0.09422 

0.09680 

0.09939 

0. IOI99 

0. 10459 

0. 10719 

38 

52 

0. 10719 

0. 10980 

0. 11241 

0. 11502 

0. 11764 

0. 12026 

0. 12289 

37 

53 

0. 12289 

0. 12552 

0. 12815 

0.13079 

0.13344 

0. 13608 

0.13874 

36 

54 

0.13874 

0. 14140 

0. 14406 

0.14673 

0. 14941 

0.15209 

0.15477 

35° 

55° 

0.15477 

0.15746 

0. 16016 

0. 16287 

0. 1655S 

0. 16829 

0. 17101 

34 

56 

0. 17101 

0.17374 

0. 17648 

0. 17922 

0. 18197 

0. 18472 

0.18748 

33 

57 

0.18748 

0. 19025 

0.19303 

0. 19581 

0. 19860 

0. 20140 

0 . 20421 

32 

58 

0. 20421 

0.20703 

2. 209S5 

0.21268 

0.21552 

0.21837 

0. 22123 

31 

59 

0.22123 

0. 22409 

0. 22697 

0. 229S5 

0.23275 

0.23565 

0.23856 

30° 

60° 

0.23S56 

0.24148 

0.24442 

0.24736 

0.25031 

0.25327 

0.25625 

29 

61 

0.25625 

0.25923 

0. 26223 

0. 26524 

0.26825 

0.27128 

0-27433 

28 

62 

0-27433 

0.27738 

0. 28045 

0.2S352 

0.28661 

0.28972 

0.29283 

27 

63 

0.292S3 

0. 29596 

0.29911 

0. 30226 

0.30543 

0.30862 

0. 31 182 

26 

64 

0. 31182 

0.31503 

0.31826 

0.32150 

0.32476 

0.32804 

0.33133 

25° 

65° 

o-33^33 

0.33463 

0.33796 

0.34130 

0.34465 

0.34803 

0.35142 

24 

66 

0.35142 

0.35483 

0.35825 

0.36170 

0.36516 

0.36865 

0-37215 

23 

67 

0.37215 

0.37567 

0.37921 

0.38278 

0.38636 

0.38996 

0-39359 

22 

68 

0-39359 

0.39724 

0.40091 

0. 40460 

0.40832 

0.41206 

0. 41582 

21 

69 

0.41582 

0.41961 

0.42342 

0.42726 

0.43II3 

0.43502 

0.43893 

20° 

70° 

0.43893 

0.44288 

0.44685 

0.45085 

0. 45488 

0.45894 

0.46303 

19 

71 

0.46303 

0.46715 

0.47130 

0.47548 

0.47969 

0.48394 

0.48822 

18 

72 

0.48822 

0.49254 

0.49689 

0.50128 

0.50570 

0. 51016 

0.51466 

17 

73 

0.51466 

0. 51920 

0.52378 

0. 52840 

0.53306 

0.53776 

0.54250 

16 

74 

0.54250 

0.54729 

0.55213 

0.55701 

0.56194 

0. 56692 

0.57195 

15° 

75° 

0.57195 

0.57703 

0.58216 

0.58734 

0.59258 

0.59788 

0.60323 

14 

76 

0.60323 

0. 60864 

0. 61411 

0. 61965 

0.62524 

0.63091 

0.63664 

13 

77 

0.63664 

0.64243 

0. 64S30 

0.65424 

0.66026 

0.66635 

0.67253 

12 

78 

0.67253 

0.67878 

0.68511 

0. 69154 

0.69805 

0.70465 

0-71135 

II 

79 

0.7113s 

0. 71814 

0.72504 

0.73203 

0.73914 

0.74635 

0.75368 

10° 

80° 

0.75368 

0.76113 

0.76S70 

0.77639 

0.7S422 

0. 79218 

0.80029 

9 

81 

0.80029 

0.80854 

0. 81694 

0. S2550 

0.83423 

0.84312 

0.85220 

8 

82 

0.85220 

0.86146 

0. 87091 

0.S8057 

0. 89044 

0.90053 

0.91086 

7 

83 

0.91086 

0.92142 

0.93225 

0.94334 

0.95472 

0.96639 

0.97838 

6 

84 

0.97838 

0.99070 

1.00338 

I .01642 

1.02987 

I -043 73 

1.05805 

5° 

85° 

1.05805 

1.07284 

1.08S15 

1. 10402 

1. 12047 

I. 13757 

1-15536 

4 

86' 

1-15536 

1.17390 

I. 19326 

1.21351 

1-23475 

1.2570S 

I . 28060 

3 

87 

I  .^28060 

1.30547 

I. 33184 

1.35991 

1.38991 

1 .42212 

1.45692 

2 

88 

1.45692 

r. 49473 

1.53615 

1.58193 

I. 63311 

1.69112 

1.75808 

I 

89 

1.75808 

1-83727 

1.93419 

2.05914 

2.23524 

2/53627 

00 

°° 

60'     50'     40'    30'     20'     10'     0'  / 

Lngle 

LOG  COTANGENT 


J    l/^ 


52  Logarithmic  Tables 

33.  Logarithms  of  Trigonometric  Functions 


Angle 

1° 

Log  Arc 
2 .  2419 

Log  Sin 

Log  Tan  1  Log  Sec 

Log  Csc 

Log  Cot 

Log  Cos 

2.2419 

2. 2419 

0. 0001 

1.7581 

1.7581 

1-9999 

0.1913   89 

3 

2.5429 

2.5428 

2.5431 

0.0003 

1-4572 

1.4569 

1-9997 

0.1864 

88 

3 

2. 7190 

2.7188 

2.7194 

0.0006 

1.2812 

1.2806 

1-9994 

0. 1814 

87 

4 

2.8439 

2.8436 

2  .  8446 ' O.OOII 

1.1564 

1-1554 

1 . 99S9 

0.1764 

86 

5 

2 . 9408 

2 . 9403 

2.9420  0.0017 

1-0597 

1.0580 

1-9983 

0.1713 

85° 

6° 

1.0200 

I. 0192 

T. 0216 

0.0024 

0.9808 

0.9784 

1-9976 

0. 1662 

84 

7 

1.0870 

1.0859 

T. 0891 

0. 0032 

0.9141 

0. 9109 

1.9968 

0.1610   83 

8 

1.145° 

T.1436 

T.1478 

0. 0042 

0.8564 

0. 8522 

1.9958 

0.1557 

82 

9 

T. 1961 

I. 1943 

I . 1997 ' 0. 0054 

0.8057 

0.8003 

1.9946 

0.1504 

81 

lO 

T.2419 

1-2397 

1.2463 

0. 0066 

0. 7603 

0-7537 

1-9934 

0.1450 

80° 

11° 

1.2833 

1.2806 

T.2887 

0.0081 

0.7194 

0.7113 

1-9919 

0.1395 

79 

12 

1.3211 

I-3179 

I . 3275 ' 0.0096 

0.6821 

0.6725 

1.9904 

0.1340 

78 

13 

1.3558 

I-3521 

T.3634 

0.0113 

0.6479 

0.6366 

T.9887 

0. 1284 

77 

14 

1.3880 

1-3837 

1.3968 

0.0131 

0.6163 

0. 6032 

1.9869 

0. 1227 

76 

15 

I. 4180 

1.4130 

1.4281 

0.0151 

0.5870 

0.5719 

1-9849 

0. 1169 

75° 

i6° 

T.4460 

1.4403 

1.4575 

0.0172 

0.5S97 

0.5425 

T.9828 

0. iiii 

74 

17 

1.4723 

1-4659 

1.4853 

0.0194 

0.5341 

0.5147 

T. 9806 

0. 1052 

73 

i8 

I. 4971 

T. 4900 

1.5118 

0.0218 

0. 5100 

0.4882 

1.9782 

0.0992 

72 

19 

1.5206 

I. 5126 

1.5370 

0.0243 

0.4874 

0.4630 

1-9757 

0.0931 

71 

20 

1.5429 

I -5341 

1.5611 

0.0270 

0.4659 

0.4389 

1-9730 

0.087a 

70° 

21° 

1.5641 

1-5543 

1.5842 

0.0298 

0.44S7 

0.4158 

T.9702 

0.0807 

69 

22 

I  •  5843 

1-5736 

1 . 6064 

0.0328 

0. 4264 

0.3936 

1-9672 

0.0744I  68 

23 

1 . 6036 

1-5919 

T.6279  0.0360 

0. 4081 

0.3721 

I . 9640 

0.0G80  67 

24 

1 .  6221 

1.6093 

1.6486 

0-0393 

0.3907 

0.3514 

I . 9607 

0.0614  66 

25 

1.6398 

1.6259 

T.6687 

0.0427 

0.3741 

0-3313 

1-9573 

0.0548 

6s° 

26° 

1.6569 

T.6418 

T.6882 

0.0463 

0.3582 

0.3118 

T-9537 

0. 0481 

64 

27 

T.6732 

1.6570 

1.7072 

0. 0501 

0.3430 

0. 292S 

1-9499 

0. 0412 

^3 

28 

1.6890 

1.6716 

1.7257 

0.0541 

0.3284 

0.2743 

1-9459 

0.0343 

62 

29 

T.7042 

1.6856 

1.7438 

0.0582 

0.3144 

0. 2562 

1.9418 

0.0272 

61 

30 

1.7190 

I . 6990 

1.7614 

0.0625 

0. 3010 

0.2386 

I -9375 

0.0200 

60° 

31° 

1.7332 

T.7118 

T.7788 

0.0669 

0.2882 

0. 2212 

1-9331 

0. 0127 

59 

32 

I . 7470 

1.7242 

1.7958 

0.0716 

0.2758 

0.2042 

1.92S4 

0.0053 

58 

33 

T. 7604 

1.-361 

1.8125 

0.0764 

0.2639 

0.1875 

1.9236 

1-9978 

57 

34 

1-7734 

1.7476 

T. 8290 

0. 0814 

0.2524 

0. 1710 

T.91S6 

I .9901 

56 

35 

1-7859 

T.7586 

T.8452 

0.0S66 

0. 2414 

0.1548 

1-9134 

T.9822 

55° 

36° 

1.7982 

T.7692 

T.8613 

0. 0920 

0. 2308 

0.1387 

1.90S0 

1.9743 

54 

37 

1.8101 

1.7795 

T.8771 

0.0977 

0. 2205 

0. 1229 

1-9023 

I .9G62 

53 

38 

I. 8217 

1-7893 

T.8928 

0.1035 

0. 2107 

0. 1072 

1-8965 

I -9579 

52 

39 

1.8329 

1-7989 

T.90S4 

0. 1095 

0.2011 

0. 0916 

1.8905 

1-9494 

51 

40 

I. -8439 

T.8081 

1.9238 

0.1157 

0. 1919 

0.0762 

1-8843 

X . 9408 

50° 

41° 

1-8547 

T.8169 

1-9392 

0. 1222 

0.1831 

0.0608 

1-8778 

1-9321 

49 

42 

T.8651 

1-8255 

1-9544 

0. 1289 

0.1745 

0.0456 

I.8711 

19231 

48 

43 

T-8753 

1-8338 

1.9697 

0.1359 

0. 1662 

0.0303 

T.8641 

T.9140 

47 

44 

1.8853 

T.8418 

T.984S 

0.1431 

0. 1582 

0. 0152 

1-8569 

I . 9046 

46 

45 

I. 895 1 

1-8495 

0.0000 

0-1505 

0.1505 

0. 0000 

1-8495 

1-8951 

45° 

Log  Cos 

Log  Cot 

Log  Csc 

Log  Sec 

Log  Tan 

Log  Sin 

Log  Arc 

Angle 

Logarithmic  Tables  53 

34.  Explanations 

Table  30  gives  five-place  Logarithms  of  three-place  numbers. 
The  word  logarithm  and  its  abbreviation  log,  when  used  without 
qualification,  refer  to  a  common  logarithm  which  is  defined  by 
the  equation  10'°^  "=n.  The  table  gives  the  decimal  part,  or 
mantissa,  of  a  logarithm,  while  the  integral  part,  or  characteristic, 
is  to  be  supplied  by  the  follo^\ing  rules:  When  the  number  is 
greater  than  1 ,  the  characteristic  of  its  log  is  positive  and  is  one 
less  than  the  number  of  figures  preceding  the  decimal  point;  thus, 

log  6.54=0.81558       log  65.4  =1.81558       log  654  =2.81858  • 

When  the  number  is  less  than  1,  the  characteristic  of  its  log  is 
negative  and  is  numerically  one  greater  than  the  number  of  ciphers 
immediately  following  the  decimal  point,  thus  the  four-place 
log  of  6  is  0.77S2,  and 

log  0.6  =T.77S2       log  0.06  =  2.7782       log  0.006  =3.7782 

Here  the  characteristic  is  negative  and  the  mantissa  is  positive,  so 
that  2.7782  is  the  same  as  —2+0.7782.     When  the  given  number 
is  an  integral  power  of  10,  the  mantissa  is  zero,  so  that  log  1000 
=3,  log  0.1  =  - 1,  log  0.01  =  -2,  and  log  1  =0. 

Multiplication  and  Division  may  be  performed  by  the  help  of 
logarithms  and  the  use  of  the  following  rules: 

To  multiply  a  by  6,  log  a+log  b  =log  ab 

To  divide  a  by  b,  log  a  —log  b  =log  a/b 

Here  log  a  and  log  b  are  obtained  from  Table  30  and  the  above  rules 
for  the  characteristic;  then  the  numbers  corresponding  to  log  cb 
and  log  a/b  are  found  from  the  Table.  For  example,  to  mAil- 
tiply  68.31  by  0.2754,  the  sum  of  the  logs  is  1.27444  and  its  corre- 
sponding number  is  18.812,  the  last  decim.al  being  in  error. 

Roots  and  Powers  of  numbers  are  m.ost  conveniently  computed 
by  logarithms  and  the  use  of  the  following  rules: 

To  extract  the  nth  root  of  a,  -log  a  =log  a'* 

n 

To  raise  a  to  the  mth  power,        m  log  a  =Iog  a^ 
For  example,  to  raise  0.8831  to  the  1.53  power:   1.53X1.83448 


52  Logarithmic  Tables 

33.  Logarithms  of  Trigonometric  Functions 


Angle 
1° 

'Log  Arc 
2. 2419 

Log  Sin 

Log  Tan  1  Log  Sec 

Log  Csc 

Log  Cot 

Log  Cos 

2.2419 

1 
2.2419  O.OOOI 

1.7581 

1.7581 

1.9999 

0.1913'  89 

2 

2.5429 

2.5428 

2.5431  0.0003 

1-4572 

1.4569 

1.9997  0.1864  88 

3 

2.7190 

2.7188 

2.7194  0.0006  I. 2812 

1 . 2806 

1-9994 

0.1814   87 

4 

2.8439 

2.8436 

2.8446  o.ooii  I. 1564 

1.1554 

1.99S9 

0.1764  86 

5 

2 . 9408 

2 . 9403 

2.9420 

0. 0017 

1-0597 

1 .05S0 

1 ■ 9983 

0.1713,  85° 

6° 

1.0200 

T.0192 

I. 0216 

0.0024 

0.9808 

0.9784 

1-9976 

0. 1662 

84 

7 

T.0870 

1.0859 

I . 0891 

0.0032 

0.9141 

0.9109 

1.996S 

0.1610'  83 

8 

T.1450 

I. 1436 

I. 1478 

0.0042 

0.S564 

0.8522 

1-995S 

0.1557,  82 

9 

I. 1961 

I. 1943 

I. 1997 

0. 0054 

0.8057 

0.8003 

1.9946 

0.1504 

81 

lO 

I. 2419 

1.2397 

I . 2463 

0.0066 

0.7603 

0.7537 

1-9934 

0. 1450 

80° 

11° 

1.2833 

1.2806 

T.2887  0.0081 

0.7194 

0.7113 

1-9919 

0.1395 

79 

12 

T.3211 

I. 3179 

1.3275  0.0096 

0.6821 

0.6725 

1 . 9904 

0.1340 

78 

13 

1.3558 

I. 3521 

T. 3634  0.0113 

0.6479 

0.6366 

T.9887 

0. 1284 

77 

14 

T.3880 

1.3837 

i.3968j0.oi3i 

0. 6163 

0.6032 

1.9869 

0. 1227 

76 

15 

T.4180 

T.4130 

1.4281 

0.0151 

0.5870 

0.5719 

1.9849 

0.1169J  75° 

i6° 

T .  4460 

1.4403 

1.4575 

0.0172 

0.5597 

0.5425 

1.9828 

0. iiii 

74 

17 

1-4723 

1.4659 

1-4853 

o.oi94;o.534i 

0.5147 

T.9806 

0. 1052 

73 

i8 

I. 4971 

T.4900 

1.51181  o.o2i8|o.5ioo 

0.4882 

1.9782 

0.0992 

72 

19 

T. 5206 

I. 5126 

T-5370 

0.0243  0.4874 

0.4630 

1.9757 

0.0931 

71 

20 

1.5429 

I. 5341 

1.5611 

0.0270  0.4659 

0.4389 

1.9730 

0.087QJ  70° 

21° 

T.5641 

1.5543 

1.5842 

0.0298I0.4457 

0.4158 

T.9702 

0.0807 

1  69 

22 

15843 

1.5736 

I . 6064 

0.0328 

0.4264 

0.3936 

T.9672 

0.0744  68 

23 

1 .  6036 

I. 5919 

T. 6279  0.0360 

0.4081 

0.3721 

1 . 9640 

0.0680  67 

24 

T. 6221 

1.6093 

T.6486 

0.0393 

0.3907 

0.3514 

T.9607 

0.0614  66 

25 

1.6398 

1.6259 

1.6687 

0.0427 

0.3741 

0.3313 

1-9573 

0.0548  65° 

26° 

1.6569 

T.6418 

1.6S82 

0.0463 

0.3582 

0.3118 

T-9537 

0.0481   64 

27 

1.6732 

1.6570 

1.7072 

0.0501 

0-3430 

0. 2928 

1.9499 

0.0412  03 

28 

1.6890 

I. 6716 

1-7257 

0.0541 

0.3284 

0.2743 

1-9459 

0.0343 

62 

29 

T.7042 

T.6856 

I  -  7438 

0.0582 

0.3144 

0.2562 

I. 9418 

0.0272 

61 

30 

I. 7190 

T.6990 

I. 7614 

0.0625 

0.3010 

0.2386 

1-9375 

0.0200 

60° 

31° 

1.7332 

T.7118 

1.7788 

0.0669 

0.2882 

0.2212 

1-9331 

0. 0127 

59 

32 

T.7470 

1.7242 

1.7958 

0.0716 

0.2758 

0.2042 

1.9284 

0.0053 

58 

33 

1 .  7604 

I. -361 

1.8125 

0.0764 

0.2639 

0.1875 

1-9236 

1.9978 

57 

34 

1-7734 

1.7476 

T.8290 

0.0814 

0.2524 

0. 1710 

1.9 1 86 

T.9901 

56 

35 

1.7859 

T.75S6 

1.8452 

0.0S66 

0.2414 

0. 1548 

1-9134 

T.9S22 

55° 

36° 

T.7982 

T.7692 

I. 8613 

0.0920 

0.2308 

0.1387 

T.9080 

1-9743 

54 

37 

1.8101 

1-7795 

T.8771 

0.0977 

0.2205 

0. 1229 

1.9023 

I .9662 

53 

38 

T.8217 

1.7893 

T.8928 

0.1035 

0. 2107 

0. 1072 

1-8965 

1-9579 

52 

39 

T.8329 

1-7989 

1.90S4 

0. 1095 

0. 2011 

0.0916 

1  -  8905 

1.9494 

51 

40 

1.8439 

1.80S1 

T.9238 

0-1157 

0. 1919 

0.0762 

1.8843 

1.9408 

50° 

41° 

1-8547 

T.8169 

1.9392 

0. 1222 

0.1831 

0.0608 

T.877S 

T.9321 

49 

42 

T.86S1 

1-8255 

1.9544 

0. 1289 

0.1745 

0.0456 

T.8711 

1. 923 1 

48 

43 

7-8753 

1-833S 

1.9697 

0.1359 

0. 1662 

0.0303 

T.8641 

I. 9 140 

47 

44 

T-S853 

T.8418 

T.9848 

0.1431 

0. 1582 

0.0152 

T.8569 

T.9046 

46 

45 

I.  895 1 

1-8495 

0 . 0000 

0-1505 

0.1505 

0.0000 

1-8495 

1.8951 

45° 
Angle 

L.og  Cos 

-og  Cot 

Log  Csc 

Log  See, 

Log  Tan 

Log  Sin 

Log  Arc 

Logarithmic  Tables  53 

34.  Explanations 

Table  30  gives  five-place  Logarithms  of  three-place  numbers. 
The  word  logarithm  and  its  abbreviation  log,  when  used  without 
qualification,  refer  to  a  common  logarithm  which  is  defined  by 
the  equation  10'°^  "=n.  The  table  gives  the  decimal  part,  or 
mantissa,  of  a  logarithm,  while  the  integral  part,  or  characteristic, 
is  to  be  supplied  by  the  follo^ving  rules:  When  the  number  is 
greater  than  1,  the  characteristic  of  its  log  is  positive  and  is  one 
less  than  the  number  of  figures  preceding  the  decimal  point ;  thus, 

log  6.54  =0.81558       log  65.4  =  L81558       log  654  =2.81858 

When  the  number  is  less  than  1,  the  characteristic  of  its  log  is 
negative  and  is  numerically  one  greater  than  the  number  of  ciphers 
immediately  follo^^■ing  the  decimal  point,  thus  the  four-place 
log  of  6  is  0.7782,  and 

log  0.6  =T.77S2      log  0.06  =  2.7782      log  0.006  =3.7782 

Here  the  characteristic  is  negative  and  the  mantissa  is  positive,  so 
that  2.7782  is  the  same  as  —2+0.7782.     When  the  given  number 
is  an  integral  power  of  10,  the  mantissa  is  zero,  so  that  log  lOCO 
=  3,  log  0.1=  -1,  log  0.01  = -2,  and  log  1=0. 

^Multiplication  and  Division  may  be  performed  by  the  help  of 
logarithms  and  the  use  of  the  following  rules : 

To  multiply  a  by  b,  log  «+log  b  =log  ab 

To  divide  a  bj'  b,  log  a  —log  b  =log  a/b 

Here  log  a  and  log  b  are  obtained  from  Table  30  and  the  above  rules 
for  the  characteristic;  then  the  numbers  corresponding  to  log  ab 
and  log  a/b  are  found  from  the  Table.  For  example,  to  m.ul- 
tiply  68.31  by  0.2754,  the  sum  of  the  logs  is  1.27444  and  its  corre- 
sponding number  is  18.812,  the  last  decim.al  being  in  error. 

Roots  and  Powers  of  numbers  are  most  conveniently  computed 
by  logarithms  and  the  use  of  the  following  rules : 

To  extract  the  nth  root  of  a,  -log  a  =log  a" 

n 

To  raise  a  to  the  mth.  power,        m  log  a  =log  a^ 
For  example,  to  raise  0.G831  to  the  1.53  power:    1.53X1.83448 


54  Logarithmic  Tables 

=  -1.53+1.27675  =  2.47+1.27675=1.74675,  which  is  log  of 
0.55815.  To  find  the  fifth  root  of  0.6831:  one-fifth  of  I.S3448 
is  i  (-5+4.83448)  =1.96690,  which  is  log  of  0.9262;  or  it  is  per- 
haps better  to  multiply  bj'  0.2  instead  of  diWding  by  5,  thus  0.2 
(I.S3448)  =  0.2  (-  1  +  0.83448)  =  -  0.2  +  0.16690  =  -1  +  0.8 
+0.16690=1.96690. 

Tables  31  and  32  give  logarithms  of  trigonometric  functions  to 
five  decimal  places  at  intervals  of  10',  the  characteristics  being 
given.  For  log  sin  and  log  tan  look  for  the  degree  at  the  left-hand 
side  and  for  the  minutes  at  the  top;  for  log  cos  and  log  cot  look 
for  the  degree  at  the  right-hand  side  and  for  the  minutes  at  the 
foot.  In  many  books  these  functions  arc  called  logarithmic  sines, 
logarithmic  tangents,  etc.,  while  the  characteristics  are  WTitten 
8  and  9  instead  of  2  and  T,  thus  requiring  some  power  of  10  to 
be  subtracted  later.  Here  the  final  logarithm  of  a  computation  is 
correct  without  such  subtraction. 

Table  33  gives  four-place  logarithms  of  trigonometric  func- 
tions, and  its  arrangement  is  the  same  as  that  of  Table  29. 

35.  Exercises 

1.  Find  the  logarithms  of  7.25,  7250,  and  0.725. 

2.  Find  the  nuinbors  whose  logarithms  are  1.64933,  G.64933,  2.64933, 
0.70520,  1.70520,  and  0.73998. 

3.  Compute  by  logarithms  the  sixth  powers  of  3.25  and  0.325; 
also  the  sixth  roots  of  3.27  and  0.327. 

4.  Using  logarithms,  multiply  32. IG  by  0.01555;  also  divide  1825 
by  0.03245. 

5.  Find  log  sinos  of  44°  22'  and  44°  25';  also  log  cosines  and  tan- 
gents of  the  same  angles. 

0.  r.ivcn  a  =  h  sin  A  /sin  B  compute  the  value  of  a  when  !>  =973  feet, 
^=24°40',  and/^  =  73°  10'. 

7.  Compute  the  value  of  0.375  tan  85°;  also  of  sec  7S°Xcos7S°; 
also  of  cot  39°  lO'Xsin  39°  lO'/cos  39°  10'. 

8.  In  a  right-angled  triangle  the  hypothcnuse  is  505  feet;  compute 
the  other  two  sides  when  one  of  the  acute  angles  is  53°  8'. 

9.  When  a  vortical  post  3.125  feet  high  casts  a  shadow  8.275  feet 
long  on  a  level  plane,  what  is  the  elevation  of  the  svui  above  the  horizon? 


Chapter  G 
WEIGHTS  AND  MEASURES 


56 


Weights  and  Measures 


36.  Length 

I  meter  =  10  decimeters  =ioo  centimeters  =  looo  millimeters=io' microns  =o.i  deka- 
meter=o.oi  hectometer  =  o.ooi  kilometer  =  o-oooi  myriameter. 
I  U-  S.  yard  =  3600/3937  meters  (by  definition);  log  =  i.96ii37i. 


Meters 

Inches 

Feet 

Yards 

Links 

R°d^-       Chains. 

P°'^^'°^   Gunter's 
perches 

Statute 
miles 
U.S. 

Nautical 
miles 
U.S. 

I 

39-37 

1.59517* 

3.2808 
0.51598* 

1.0936 
0.03886* 

4-971 
0.69644* 

0.1988      0.04971 
1.29850*  2.69644* 

0.(3)6214 
4-79335* 

0.(3)5396 
4.73207' 

0.0254 
5.40483* 

I 

0.08333 
2.920S2* 

■0.02778 
2.44370* 

0.1263 
1.10127* 

0.0050S1 
3-70333* 

0.001263 
3.10127* 

0.(4)1578 
SJ9818* 

o.(4)i37l 
S-13690 

0.3048 
1.48402* 

12 

I. 07918* 

1 

0.3333 
1.52288* 

i-SiS 
0.18046* 

0.06061 
2.78252* 

o.oisis 
2. 18046* 

0.(3)1894 
4-27737* 

o.(3)i6dS 
4.21608' 

0.9144 
1.96114* 

36 

I -55630* 

3 

0.47712* 

I 

4-545 
0-65758* 

0.1818 
1-25964* 

0.0454s 
2.65758* 

0.(3)5682 
4-75449* 

o.(3)4934 
4.69320* 

0.2012 
1.30356* 

7-92 

0.89S73* 

0-66 

I-81954* 

0.22 

1.34242* 

I 

O-04 

2-60206* 

O.OI 

2.00000* 

0.(3)1250 
4.09691* 

o.(3)lo86 
4-03564 

5.029 
0.70150* 

198 

2. 29667* 

16-5 

1. 2 1 748* 

5-5 

0. 74036* 

25 

1.39794* 

I 

0.25 

1-39794* 

o.(2)3I2S 
3.49485* 

0.(2)2714 
3-43357* 

20- 12 
1.30356* 

792 

2.89S73* 

66 

1.81954* 

22 

1.34242* 

100 

2-00000* 

4 
0-60206* 

1 

0.0125 

2.09691* 

0.010S6 
2.03564* 

1609.3 
3.2066s* 

63360 

4.801S2* 

5280 

3.72263* 

1760 

3-245SI* 

8000 

3-90309* 

320 

2.50515* 

80 

1.90309* 

1 

0. 8684 
1.93873 

1853-25 
3.26753* 

72962 
4.86310* 

6080. 2 

3.78392* 

2026.73 
3-30680* 

9212 
3-96437* 

368- 5 
2-56643* 

92.12 
1.96437* 

1. 1516 
0.06128* 

I 

I  nautical   mile  of   the  British  admiralty  =  6080  ft.     i   furlong  =  ^3  mile  =  660  feet. 
1  league  =  3  miles  =  24  furlongs.     1  fatliom  =  2  yards  =  6  feet. 
*  Logarithm  of  the  number  immediately  above. 


37.  Area 
I  hectare  =  100  ares  =  10  000  centares  or  square  meters. 


Square 
meters 

Square 
inches 

Square 
feet 

Square 
yards 

Square 
rods 

Square 
chains 

.\cres 

Square 
miles  or 
sections 

1 

1550 
3-19033* 

10-764 
1-03197* 

1. i960 
0.07773* 

0-03954 
2.59700* 

0-12)2471 
3 -39  288* 

0.(3)2471 
4.3928S* 

0.  (6)3861 
7-  38670' 

o.(3)54S3 
4.80967* 

I 

0-006944 
3.84164* 

o.C2)77i6 
3.88740* 

0  (4)2551 
5-40667* 

0.(0)1594 
6-20255' 

0.(6)1594 
7.20255' 

0-  (9)2491 
10.39637* 

0.09290 
2.96803* 

144 

2.15S36* 

I 

O.IllI 

1.04576* 

0.(2)3673 
3.56503* 

0.(3)2296 
4-36091* 

0.(4)2296 
S. 3609 1* 

0.  (7)3587 
8.  55473* 

0.8361 
1.92227* 

1296 
3- 1 1 260* 

9 

0.95424* 

I 

0-03306 
2-51927* 

o.(2)2o66 
3-31515* 

o.(3)2o66 
4.31315* 

0.  (6)3228 
7.  S0898' 

25.29 
1.40300* 

39204 

4-5933* 

272.25 

2-43497* 

30-23 

1-48072* 

I 

0.0625 

2.7958S* 

0.00625 

3-79588* 

0.  (5)9766 
6.98970' 

404.69 
3.60712* 

627264 

5-79745* 

4356 

3-63909* 

484 

2.6S484* 

16 

1-20412* 

1 

O-I 

I- 00000* 

0.  (3)IS62 

4.19382* 

4046.9 
3.60712* 

6272640 

6-79745* 

43560 

4.63909* 

4840 

3.6S4S4* 

160 

2-  20.(I2* 

10 

1 . 00000* 

1 

0.001562 
3.19382* 

2589998 
6. 41330* 

27878400 

7-44527* 

3097600 

6.49102* 

102400 

5.01030* 

6400 

3.80618* 

640 

3.80618* 

I 

*  Logarithm  of  the  number  immediately  above. 


Weights  and  Me.^sures 


57 


38.  Speed 

and  Velocity 

Cm  per 
sec 

Km  per 
hour 

Ft  per 

sec 

Ft  per 
min 

Miles  per 
hour 

Knots 

I 

0.036 

2-53630* 

0.032SI 
2.51598* 

1.96S5 
0.29413* 

0.02237 
2.34965* 

0.01942 
2.2S825* 

27-777S 
1.44370* 

1 

0.9II34 
T.9596S* 

54.6806 

1.737S3* 

0.62137 

1-79335* 

0.53960 
1.73207* 

30.4801 
1.48402* 

I-0973 
0.04032* 

1 

60 

1-77S1S* 

0.6S182 
T.83367* 

0.59209 
1-77238* 

0.5080 
1.70586* 

0.01829 
2.26217* 

0.01667 

2.22IS5* 

I 

0.01136 
2-05553* 

0.009868 
3-99423* 

,44-7041 
1.63035* 

1-6093 
0.20670* 

1.46667 
0.16633* 

88 

1.9444S* 

1 

0.86S39 
1.93872* 

SI. 4971 

1.71178* 

1-8332 
0.26793* 

1.68894 
0.22761* 

101.337 

2.00577* 

1-13155 
0.06128* 

I 

I  knot  =  I  nautical  mile  per  hour. 
*  Logarithm  of  the  number  immediately  above. 

39.  Voliime  and  Capacity 

I  literal  cubic  decimeter  =  1000  cubic  centimeters  =  10  deciliters  =  100  centiliters  =" 
1000  milliliters  =  0.1  dekaliter  =  0.01  hectoliter  =  0.01  kiloliter  =  0.001  cubic  meters  or 
steres. 


Cubic 
inches 

Cubic 
feet 

Cubic 
yards 

U.  S.  quarts 

Gallons 

Bushels 
U.S. 

Liters 

Liquid 

Dry 

U.S. 
liquid 

U.S. 
dry 

I 

0.(3)57870 
4.76246* 

0.  ■4^2143  0.017316 
5.33109*  2.23845* 

0.014881 
2.17263* 

0.004329 
3.63639* 

0.003720 
3-57057* 

0.(3)4650 
4-66748* 

0.016387 
2.21430* 

1728 

3-23754* 

I 

0.037037  29.922 
2.56864*11.47599* 

25-714 
1.41017* 

7-4805 
0-87393* 

6.4285 
0.8081 I* 

0.80356 
T. 90502* 

28.317 

1.45205* 

46656 

4-66891* 

27 

1-43136* 

I 

807.90      694.28 
2.90736*  2.84153* 

201.97 
2.30330* 

173-57 
2.2394S* 

21.696 
1-33638* 

764.56 
2.88341* 

57-75 
1.76155* 

0.033420 
2.52401* 

0.001238 
3.09026* 

1 

0-83937 
I. 93418* 

0.25 

1-39794* 

0.21484 
1.33212* 

0.026855 

2.42903* 

0.94636 
T. 97606* 

67.201 
1.82737* 

0.038889 
2.589S3* 

0.001440 
3-15847* 

1.1637 
0.06582* 

1 

0.29091 
1.46376* 

0.25 

1-39794* 

0.03125 

2.49485* 

I. 1012 
0.04188* 

231 

2.36361* 

0.13368 

r. 12607* 

0.004951 
3-69471* 

4 

0.60206*1 

3-4375 
0.53624* 

1 

0-85937 
T- 93418* 

0.10742 
T. 03109* 

3-7854 
0.57812* 

268.80 
2.42943* 

0-15536 
I. 19189* 

0.0037C1  4.65^6 
3.76o53*,o.66788* 

4 

0.60201* 

I. 1637 
0.06582* 

I 

0.125 

1.09691* 

4-4049 
0.64394* 

2150.4 
3-33253* 

1-2445 
0.09498* 

0.046091  37.237 
2.66362*  1.57097* 

32 

1-50515* 

9.3092 
0.96891* 

8 

0.90309* 

I 

35-239 
1-54703* 

61.023 

1.78550* 

0.035313 

2-54793* 

0.00130S  1.0567 
3.li639*|o.02394*i 

0.9080S  1 

r.9s8i2*j 

0.26417 
1.4218S* 

0.22702 

T. 35606* 

0.028377 
2-45297* 

I 

I  U.S.  liquid  quart  =  2  pLnts  =  8  gLlls  =  32  fluid  ounces  =  256  fluid  drams  =  768  fluid 
scruples,     i  bushel  =  4  piecks. 

1  Imperial  gallon  =  1.201  U.  S.  gallons  =  0.1605  cu  ft  =  4.3460  liters. 

I  U.  S.  gallon  =  0.8327  Imperial  gallons.        1  cubic  foot  =  6.229  Imperial  gallons, 

1  British  bushel  =  1.2837  cubic  feet. 

Shipping  Measure:  i  register  ton  =  100  cu  ft.        i  U.  S.  shipping  ton  =  40  cuft> 
I  British  shipping  ton  =  42  cu  ft 

•  Logarithm  of  the  number  immediately  above. 


58 


Weights  and  Measures 


40.  Weight  (Engineers'  System)  or  Mass  (Physicists'  System) 

I  kilogram  =  iooo  grams  =  o.ooi  metric  ton.     i  gm=io  decigrams=  too  centigTams  = 
looo  milligrams  =  o. I  dekagram  =  o.oi  hectogram  =  o.ooi   kilogram  =  0.0001  myri  igrara. 
1  U.S.  Avoirdupois  pound  =  04535924277  kg  =  (by  definition)  7000/5760  troy  pounds. 


Kilo- 
grams 

Grains 

Ounces 

Pounds 

Tons 

Avoir. 

Troy  and 
apoth. 

Troy  and 
apoth. 

Avoir. 

Short, 
2000  lb 

Long, 
2240  lb 

Metric, 

1000  kg 

I 

15432. 
4. 18843* 

35-274 
I-S4745* 

32-151 
1.50719* 

2.6792 
0.42801* 

2 . 2046 I 
0. 34333* 

0.001102 
3.04230* 

0.(3)9842 
4-99309* 

O.OOI 

3.00000* 

0.W6480 

S-81157* 

I 

0.(2)2286 
3-3S902* 

0.002083 
3-31876* 

0.(3)1736 

4-23958* 

o.(3)i42 

4-15490* 

0.028349 
2-45255* 

437-5 

2.6409S* 

I 

0.91146 
T-95974* 

0-075955 
2.88056* 

0.0625 

2.79588* 

0.(4)3"S 

5-49485* 

0.(4)2790 
S.44563* 

0.  (4)  283s 
5-45255* 

0.031103 
2.49281* 

480 

2.68124* 

I. 0971 
0.04026* 

I 

0.083333 
2.92082* 

o.o6857i'o.  (4)3429 
2.83614*  5-53511* 

o.(3)3o6i 
4.56508' 

0.(4)3110 
S.49281* 

0-37324 
1.57199* 

5760 

3.76042* 

13. 166 
1.11944* 

12 

I. 07918* 

1 

0.(3)4114 
4.61429* 

0.(3)3673 
4-56508* 

0.(3)3732 
4-57199* 

o-4S3i59 
1.65667* 

7000 

3.84510* 

16 

I. 20412* 

14.583 
1.16386* 

I-2IS3 
0.08468* 

I 

0.0005 

4.69897* 

0.(3)4464 
4.6497s* 

o.(3)4S36 
4-65667* 

907.18 

2-9577°* 

32000 

4-S0515* 

29167. 
4-46489* 

2430.6 
3-38570* 

2000 

3.30103* 

I 

0.892S6 
1.95078* 

0.90718 
1.95770* 

1016. I 
3.00691* 

35840 

4-55437* 

32667 
4.51410* 

2722.2 
3.43492* 

2240 

3-35025* 

I. 12 

0.04922* 

I 

I . 0 I 60 
0.00691* 

1000 

3.00000* 

35274 
4-54745* 

32151 
4.50719* 

2679.2 
3.42801* 

2204.6 
3-34333* 

1.1023 
0.04230* 

0.98421 
1.99309* 

I 

I  quarter  =  28  lb  avoir,  i  pennyweight  =  24  gr=o.o5  oz  troy.  1  oz  avoir.  =  16  drama 
avoir.  =437.5  gr.  1  stone  =  14  pounds.  1  centil  =  100  pounds.  1  hundredweight™ 
T12  pounds.     I  apothecaries'  ounce  =  8  apoth.  draras  =  24  scruples  =  48o  grains. 

♦  Logarithm  of  the  number  immediately  above. 

41.  Energy  or  Work 


Joules 
=  10'  erg 

Meter- 
kilograms 

Foot-pflunds 

Kilowatt- 
hours 

Chcval- 

vapeur- 

hours 

Horse- 
power- 
hours 

Rritish 

thermal 

units 

o-(3)947S 
4.97660* 

I 

0. 10197 

1.00848* 

0-73756 
T.S6780* 

0.(6)27778 
7-4437°' 

2-(6)37767 
7.57711* 

2-(^)3725l 
7-S7II3* 

9.80665 

0.9915207* 

I 

7-2330 
0.85932* 

0.(5)27241 

6-43522* 

0.(5)37937 
6.56863* 

0.(5)36530 
6-56265* 

0.009292 
3.96812* 

1-3558 
0.13220* 

0.13826 
I. 14068* 

I 

0.(6)37662 
7-S7S90* 

0.(6)51206 
7.70932* 

0.(6)50505 
7.70333* 

0.001285 
3.108S0* 

3.6x10" 
6-55630* 

3.6710  X  10' 
S-56478* 

2.6552X10 
6.42410* 

I 

1-3596 
0.13342* 

1.3410 
0.12743* 

3411- 
3.53290* 

2.6478X10® 
6.42288* 

270000 

S-43136* 

1-9529X10® 
6.29068* 

0.73550 
T. 86658* 

I 

0.98631 
I.9940I* 

2509. 
3.39948* 

2.6845X10" 
6.42887* 

2-7375  Xio'' 
5-43735* 

1.98x10" 

6.29667* 

0-74571 
T-87257* 

1.0139 
0.00598* 

I 

2544. 
3.40547 

to5S- 
5.02340* 

107.6 
2.03188* 

778.4 
2 . 8y 1 2  0* 

0.0  293a 
4.46710* 

o.o:>3986 
4.60051* 

0.033931 
4-59453* 

1 

Logarithm  of  the  namber  immediately  above. 


Weights  and  Measures 
42.  Pressure 


59 


KUo- 

grams 

per  sq 

cm 

Pounds 

Short 
tons, 

Atmos- 

Columns of 
mercuryt 

Columns  of  watert 

Per  sq 
in 

Persq 
ft 

persq 
ft 

pheres 

Meters 

Inches 

Meters 

Feet 

I 

14-223 
1.15300* 

2048.2 
3-3II37* 

I. 0241 
0.01034* 

0.96781 
1.98579* 

0.73553 
1.86660* 

28.958 
I. 46177* 

I 0 . 009 
1.00038* 

32.837 
I. 5 1636* 

0.070307 
2.84700* 

I 

144 

2.15836* 

0.072 

2.85733* 

0.06804 
2.83279* 

0.05 1 71 3 
2.71360* 

2.0359 
0.30876* 

0.70368 
T.8473S* 

2.3087 
0.36336* 

0.(8)4882 
4.68863* 

0.006944 
3.84164* 

I 

0.0005 

4.69897* 

0.  (3)4725 
4.67442* 

0.(3)3591 

4-55524* 

0.014138 
2.15040* 

0.004887 
3.68901* 

0.016032 
2.20500* 

0.97648 
T. 98966* 

13-889 
I. 14267* 

2000 

3-30103* 

I 

0.94504 
1-97545* 

0.71823 
1.85627* 

28.277 

1-45143* 

9-7734 
0.99004* 

32.06s 
1.50603* 

I-0333 
0.01421* 

14.697 
I. 16722* 

2116.3 
3-32558* 

T.0582 
0.02955* 

I 

O-76 

T.88081* 

29.921 
1.47598* 

10.342 
1.01439* 

33-929 
1.53058* 

1.3596 
0.13340* 

19-338 
1.28640* 

2784.6 
3.44476* 

1-3923 
0.14373* 

I. 3158 
0. 11919* 

I 

39.37 

1.59517* 

13.607 

1.13378* 

44.644 
1.64976* 

0.034S33 
3.53823* 

0.49118 
I. 69124* 

70.729 
I . 84960* 

0.035365 
2.54857* 

0.033421 
2.52402* 

0.025400 
2.40484* 

I 

0-34563 
T. 53861* 

1.1340 
0.05460* 

0.099913 
a. 99962* 

1.4211 
0. 15262* 

204.64 
2.31099* 

0.10232 
T. 00996* 

0.096697 
2.98541* 

0.073489 
2.86622* 

2.8933 
0.46139* 

I 

3.2808 
0.51598* 

0.030453 
3.48364* 

0.43315 
T. 63664* 

62.374 
1.79500* 

0.031187 
2-49397* 

0.029473 
2.46942* 

0.022399 
2.35024* 

0.88187 
T.  94540* 

0.30480 
T. 48402* 

I 

*  Logarithm  of  the  number  immediately  above. 


t  At  15°  C.  and5  =  fo. 


43.  Power 

I  kilowatt  =  1000  watts  =  1000  joules  per  second. 

1  horse-power  =  550  foot-pounds  per  second. 
I  cheval-vapeur  =  75  kilogram-meters  per  second. 


Kilowatts 

Cheval- 
vapeur 

Poncelet 

Horse- 
power 

M-kg 
per  sec 

Ft-lb 
per  sec 

Kg  cal 
per  sec 

Btu 
per  sec 

I 

1 . 3600 
0.13341* 

1.0197 

0.00848* 

1. 341 

0.12743* 

101.97 
2.00848* 

737-5 
2.86780* 

0.2388 
i. 37803* 

0.9475 
1.97660* 

0.7355 
1.86659* 

I 

0-75 

T. 87506* 

0.9863 
1.99402* 

75 

1.87506* 

542. 5 
2.73438* 

0.1756 
T. 24456* 

0.6969 
1-84318 

0.980665 

I. 99152* 

1-333 
0.12493* 

I 

1.3151 
0.11896* 

100 

2.00000* 

723-3 
2.85932* 

0-2342 
I. 36951* 

0.9292 
1-96812* 

0.7457 
r. 87257* 

1.0139 
0.00598* 

0. 7604 
T.S8104* 

I 

76.04 

I. 88104* 

550 

2.74036* 

0. 1780 
T-25055* 

0-7066 
I- 84916* 

0.009807 
3-99152* 

0-01333 
2.12493* 

O.OI 

2.00000* 

0.01315 
2.11896* 

1 

7.233 
0.85932* 

0.002342 
3-36951* 

0.009292 
3-96812* 

0.001356 
3.13220* 

0.001843 
3.26562* 

0.00138 

3.I406S* 

0.001818 
3-25964* 

0.1383 
T. 14068* 

I 

0.0003237 
4.51016* 

0.001285 
3.10880* 

4.188 

0.62201* 

5-694 
0.75542* 

4.271 
0.63049* 

5.616 
0.74945* 

427-1 
2.63049* 

3089 
3.48984* 

1 

3-968 

0.59861* 

I-OS5 
0.02340* 

1-435 
0.15682* 

1.076 
0.03188* 

1.415 

0.15084* 

107.62 

2.0318S* 

778.4 
2.89120* 

0.2520 

T. 40139* 

I 

*  Logarithm  of  the  number  immediately  above. 


60  Weights  and  Measures 

44.  Explanations 

The  preceding  Tables  give  the  numerical  relations  between 
different  units  of  measure,  all  the  numbers  in  one  horizontal  line 
being  equivalents.  For  example,  in  Table  36  the  first  line 
shows  that  1  meter  is  39.37  inches,  or  3.2S0S  feet,  or  1.0936 
yards,  etc.;  also  the  seventh  line  from  the  top  shows  that  1  yard 
is  0.9144  meters,  or  36  inches,  or  3  feet,  or  4.545  Unks,  or  0.1818 
rods,  etc. 

When  the  notation  (^)  is  seen  it  means  that  (^)  is  to  be  re- 
placed by  three  ciphers;  thus  in  the  first  line  of  Table  36  an 
equivalent  of  1  meter  is  O.OOOG214  statute  miles  and  in  Table  37 
one  square  meter  is  0.0000003SG1  square  miles. 

Below  each  equivalent  is  given  its  five-place  logarithm  marked 
with  a  *.  These  are  useful  in  converting  quantities  of  one  unit 
into  those  of  another  unit.  For  example,  to  find  how  many 
feet  there  are  in  69.39  nautical  miles:  Table  36  gives  60S0.2  as 
the  number  of  feet  in  one  nautical  mile,  hence  the  required  result 
is  69.39X6080.2  which  may  be  found  l)y  ordinary  multiplication;, 
or  by  logarithms  the  log  of  69.39  is  taken  from  Table  30  while  the 
log  of  6080.2  is  found  from  Ta])le  ,36  as  3.78392;  the  addition  of 
the  two  logs  gives  5.62522  which  is  the  log  of  421910,  where  the 
fifth  significant  figure  is  liable  to  error;  hence  the  probable 
result  obtained  from  this  table  by  use  of  the  given  logarithm  is 
that  69.39  nautical  miles  are  equivalent  to  421910  ±  2.5  feet.  By 
direct  multiplication  it  is  found  that  the  number  of  feet  is 
421905. 

Numbers  in  boldface  type  jare  exact  values,  while  all  others  in 
the  body  of  a  table  are  liable  to  an  error  of  one-fourth  of  a  unit 
in  the  last  significant  figure.  The  equivalents  above  a  table  and 
many  of  those  below  it  are  also  exact  by  definition.  As  a  general 
rule  results  obtained  by  the  use  of  equivalents  or  logarithms 
taken  from  the  body  of  a  table  are  liable  to  an  error  in  the  fifth 
significant  figure,  except  when  an  equivalent  in  the  heavy  type 
is  used  directly. 

Table  40  for  measures  of  weight  applies  also  to  measures  of 


Weights  and  Measures  61 

force  when  the  engineers'  system  is  used,  since  the  unit  of  force 
is  the  force  of  gravity  which  acts  on  the  unit  of  weight  at  lati- 
tude 45°  on  the  surface  of  the  earth. 

Tables  41  and  43  contain  some  units  which  may  not  be  familiar 
to  students  who  use  this  book,  but  the  time  will  come,  if  they  enter 
on  engineering  work,  when  the  equivalents  of  these  tables  may  be 
of  great  value  to  them.  Probably  all,  however,  know  the  mean- 
ings of  energy  or  work,  of  a  horse-power  and  a  kilowatt,  and  of  a 
British  thermal  unit;  concerning  these  a  few  exercises  are  given 
below. 

45.  Exercises 

1.  By  Table  36  how  many  feet  in  one  statute  mile?  how  many 
meters  in  one  kilometer?  how  many  statute  miles  are  equivalent  to 
one  nautical  mile? 

2.  By  Table  37  how  many  square  feet  in  one  acre?  how  many 
square  meters  are  equivalent  to  one  square  inch?  how  many  acres 
are  equivalent  to  one  square  inch? 

3.  By  Table  38  how  many  feet  per  minute  are  equivalent  to  one 
mile  per  hour?  how  many  statute  miles  per  hour  are  equivalent  to  one 
knot. 

4.  By  Table  39  how  many  bushels  are  in  one  cubic  j^ard?  how  many 
liquid  gallons  are  in  one  cubic  foot?  how  many  hters  are  equivalent  to 
1000  liquid  gallons? 

5.  By  Table  40  how  many  grains  in  one  avoirdupois  pound?  how 
many  short  tons  in  one  long  ton?  how  many  kilograms  in  one  metric 
ton? 

6.  By  Table  41  how  many  kilowatt-hours  are  equivalent  to  1  foot- 
pound?   how  many  foot-pounds  in  one  British  thermal  unit? 

7.  By  Table  42  how  many  pounds  per  square  inch  are  equivalent 
to  one  kilogram  per  square  centimeter?  how  many  feet  of  water  will 
balance  the  pressure  of  one  atmosphere? 

8.  By  Table  43  how  many  foot-pounds  per  second  make  one 
horse-power?  how  many  kilowatts  are  equivalent  to  100  horse-powers? 

9.  How  many  meters  are  equivalent  to  1000  feet?  how  many 
meters  are  equivalent  to  300  yards?  how  many  kilometers  are 
equivalent  to  62.2  statute  miles? 

10.  How  many  acres  are  equivalent  to  87,120  square  feet?  How 
many  square  meters  are  equivalent  to  153,900  square  inches? 


62  Weights  and  Measures 

11.  How  many  U.  S.  liquid  gallons  are  equivalent  to  100,000 
Imperial  gallons?  how  many  liters  are  equivalent  to  624.3  cubic 
inches? 

12.  How  many  pounds  avoirdupois  are  1000  kilograms?  how  many 
long  tons  are  37.2  metric  tons? 

13.  How  many  foot-pounds  in  0.01  kilowatt-hours?  How  many 
horse-power-hours  are  equivalent  to  6040  foot-pounds? 

14.  How  many  atmospheres  will  balance  a  column  of  water  68  feet 
high?  how  many  inches  of  mercur)'^  will  balance  100  atmospheres? 

15.  How  many  foot-pounds  per  second  are  equivalent  to  100 
horse-powers?  how  many  horse-powers  are  equivalent  to  55,000  foot- 
pounds per  second? 

16.  How  many  inches  in  one  meter?  how  many  inches  in  one  centi- 
meter? how  many  pounds  avoirdupois  in  100  long  tons  and  how  many 
in  one  metric  ton? 

17.  What  is  the  definition  of  a  horse-power?  of  a  joule?  of  a  British 
thermal  unit?  of  a  kilowatt-hour?  of  a  horse-power-hour? 


Chapter  7 
MISCELLANEOUS   TABLES 


64 


Miscellaneous  Tables 
46.  Mathematical  Constants 


Symbol 

Number 

Logarithm 

Symbol 

Number 

Logarithm 

2  Tt 

3-I415927 
6.2831853 
9.4247780 

0.4971499 
0.7981799 
0.9742711 

^n 
il-^n 

1-7724539 
0.5641896 

0.2485749 
1.7514251 

4^ 

12. 5663706 

1.0992099 

n-^2 

4.4428829 

0.6476649 

5^ 

15-7079633 

I. 1961200 

^2n 

2.5066282 

0.3990899 

671 

18.8495559 

1.2753011 

Vnl2 

I. 2533141 

o.o98o^99 

7^ 

21.9911486 

1.3422479 

Va/- 

S;r 

25.1327412 

1.4002399 

0.7978844 

I. 9019401 

9;r 

28.2743339 

I. 4513924 

£ 

2.7182818 

0.4342945 

4  7r/3 

4.1S77902 

0.6220886 

£2 

7.3890568 

0.8685890 

7Zl2 

1-5707963 

0. 1961 199 

l/£ 

0.3678794 

1-5657055 

TtiA 

0.7853982 

T. 8950899 

l/£^ 

0.1353353 

I . 1314110 

nib 

0.5235988 

I. 7189986 

/' 

0.4342945 

1-6377843 

~/3° 

0. 1047198 

T. 0200286 

i/,a 

2.3025S51 

0.3622157 

;i-/i8o 

0.0174533 

2.2418774 

sin  1° 

0.0174524 

2-2418553 

I/.T 

0.3183099 

T. 5028501 

sin  i' 

0.0002909 

4.4637261 

2  In 

0.6366198 

T. 8038801 

sin  i" 

0.0000048 

6.6855749 

i8o  In 

57-2957795 

I. 7581226 

2 

2. 

0. 3010300 

joSoo In 

3437-74677 

3-5362739 

V2 

648000  /;: 

206264.806 

5-3144251 

1.4142136 

0.1505150 

-2 

9. 8696044 

0.9942997 

VV2 

0.7071068 

1.8494850 

iln^ 

0 .  1 0 1 3  2 1 2 

1.0057003 

3 

3- 

0.4771213 

n^ 

31.0062767 

I. 4914496 

^3 

1.732050S 

0.2385606 

1/7:3 

0.0322516 

2.5085504 

s/y-i 

0-5773503 

1.7614394 

47.  Decimal  Equivalents  of  Common  Fractions 


Fract. 

1/2 

Decimal  Logarithm 

Fract 

Decimal 

Logarithm 

Fract. 
1/32 

Decimal 

Logarithm 

0-5 

1.69897 

1/8 

0.125 

I. 09691 

0.03125 

2.49485 

i/3 

0.33333 

T. 52288 

3/8 

0-37S 

1.57403 

3/32 

0.09375 

2.97197 

■^b 

0.66667 

I. 82391 

5/8 

0.625 

1.79588 

5/32 

0. 15625 

1.19382 

V4 

0.25 

1-39794 

7/8 

0.875 

I. 94201 

7/32 

O.21S75 

1.3399s 

a/4 

0-75 

1.87506 

1/12 

0.08333 

2 . 92082 

9/32 

0.28125 

1.44909 

V5 

0.2 

1-30103 

5/12 

0.41667 

T. 61979 

"/32 

0.34375 

1.53624 

2/5 

0.4 

1 . 60206 

7/12 

0.58333 

T. 76592 

1-V32 

0.40625 

T.60S79 

■•V5 

0.6 

1-7781S 

11/12 

0.91667 

I. 96211 

15/32 

0.46S75 

1.67094 

^/5 

Ve 

0.8 

0. 16667 

1 . 90309 
1.22185 
T. 92082 

Vie 

0.0625 

2.79588 

17/32 

0.53125 

1.72530 

5/6 

0.83333 

a/16 

5/16 

0.1875 
0.3125 

1.27300 
1.49485 

19/32 
21/.-12 

0.59375 
0. 656''5 

1.77360 
I. 81707 

V? 

0. 14286 

1.15490 

7/18 

0.4375 

T. 64098 

23/32 

0.71875 

T. 85658 

7? 

•V7 

0.28571 
0.42857 

1-45593 
1 .63202 
T. 75696 
1-85387 
7 -9330s 

0/16 

0.5625 

T. 75012 

2,5/32 

0.78125 

T. 89279 

■V7 

6/7 

0.57143 
0.71429 
0.85714 

11/10 

"/lO 
15/10 

0.6875 
0.8125 
0.9375 

1.83727 
1.90982 
T. 97197 

27/32 
29/32 
31/32 

0.84375 
0.90625 
0.96S75 

T. 92621 

1.95725 
1. 9862 1 

Miscellaneous  Tables 
48.  Natural  Hyperbolic  Functions 


65 


u 

Sinh  u 

Cosh  u 

Tanh  u 

« 

Sinh  H 

Cosh  M 

Tanh  u 

o.oo 

0. 0000 

I . 0000 

0 . 0000 

2.25 

4.6912 

4.7966 

0.9780 

0.05 

0.0500 

1.0013 

0. 0500 

2.30 

4.9370 

5-0372 

0. 9801 

O.IO 

0. 1002 

I .0050 

0.0997 

2.35 

5-1951 

5-2905 

0.9820 

0.15 

0. 1506 

1-0113 

0. 1489 

2.40 

5.4662 

5-5569 

0.9837 

0.20 

0.2013 

I . 0201 

0.1974 

2.45 

5-75IO 

5.8373 

0.9853 

0.25 

0. 2526 

I. 0314 

0.2449 

2.50 

6.0502 

6-1323 

0.9866 

0.30 

0.3045 

I -0453 

0.2913 

2.55 

6-3645 

6.4426 

0.9879 

0.35 

0.3572 

I .0619 

0.3364 

2.60 

6.6947 

6. 7690 

0. 9890 

0.40 

0.4108 

1.0811 

0.3800 

2.65 

7.0417 

7-1123 

0. 9900 

0.45 

0.4653 

I. 1030 

0. 4219 

2.70 

7-4063 

7-473S 

0.9910 

0.50 

0.5211 

I. 1276 

0. 4621 

2.75 

7-7894 

7-8533 

0.9918 

0.5S 

0.5782 

1-1551 

0. 5005 

2.80 

8. 1919 

8.2527 

0.9q26 

0.60 

0.6367 

1-1S55 

0.5370 

2.85 

8.6150 

8.6728 

0.9933 

0.65 

0. 6967 

I. 2188 

0-5717 

2.90 

9-0596 

9. 1146 

0.9940 

0,70 

0.7586 

1-2552 

0. 6044 

2.95 

9-5268 

9-5791 

0.9945 

0.7S 

0.8223 

1.2947 

0.6352 

3-00 

10.018 

10.068 

0.9950 

0.80 

0.88S1 

1-3374 

0. 6640 

3.0s 

10.534 

10.581 

0.9955 

0.85 

0.9561 

1-3835 

0. 6911 

3.10 

II . 076 

II . 122 

0.9959 

0.90 

1.0265 

I-4331 

0.7163 

3-15 

11.647 

11.690 

0.9963 

0.95 

1-0995 

1.4862 

0.7398 

3-20 

12. 246 

12.287 

0.9967 

1. 00 

1-1752 

I -5431 

0.7616 

3-25 

12.876 

12.915 

0.9970 

1.05 

1-2539 

1.6038 

0.7818 

3-30 

13.538 

13.575 

0.9973 

1. 10 

1-3356 

1.6685 

0. 8005 

3.35 

14.234 

14.269' 

0.9976 

1-15 

1.4208 

1-7374 

0.8178 

3-40 

14.965 

14.999 

0.9978 

1.20 

1-5095 

I. 8107 

0-8337 

3 -4  .=5 

15.734 

15.766 

0.9980 

1.25 

I . 6019 

1.8884 

0.8483 

3-50 

16.543 

16.573 

0.9982 

1.30 

1.6984 

1.9709 

0.8617 

3-55 

17.392 

17.421 

0.9984 

1-35 

I. 7991 

2.0583 

0.8741 

3-6o 

18.285 

18.313 

0.9985 

1.40 

1-9043 

2.1509 

0-8854 

3.65 

19. 224 

19.250 

0.9987 

1-45 

2.0143 

2.2488 

0.8957 

3-70 

20. 211 

20.236 

0.9988 

1.50 

2.1293 

2-3524 

0. 9052 

3.75 

21.249 

21 . 272 

0. 9989 

1-55 

2. 2496 

2.4619 

0.9138 

3.80 

22.339 

22.362 

0. 9990 

1.60 

2-3756 

2-5775 

0. 9217 

3.8s 

23.486 

23.507 

0.9991 

1.65 

2-5075 

2.6995 

0. 9289 

3-90 

24. 691 

24.711 

0.9992 

1.70 

2.6456 

2.8283 

0.9354 

3-95 

25.958 

25.977 

0.9993 

1.75 

2.7904 

2.9642 

0.9414 

4.0 

27.290 

27.308 

0.9993 

1.80 

2.9422 

3.107s 

0. 9468 

4.1 

30.162 

30.178 

0.9995 

1.85 

3-1013 

3.2585 

0.9518 

4.2 

33-336 

33.351 

0.9996 

1.90 

3.2682 

3.4177 

0. 9562 

4-3 

36.843 

36.857 

0.9996 

1.95 

3-4432 

3-5855 

0.9603 

4.4 

40.719 

40.732 

0.9997 

2.00 

3.6269 

3.7622 

0.9640 

4-5 

45.003 

45.014 

0.9998 

2.05 

3-8196 

3-9483 

0.9674 

4.6 

49.737 

49.747 

0. 9998 

2.10 

4.0219 

4- 1443 

0.9705 

4-7 

54.969 

54.978 

0.9999 

2.15 

4.2342 

4.3507 

0.9732 

4.8 

60.751 

60.759 

0.9999 

2.20 

4-4571 

4.5679 

0.97S7 

4-9 

67.141 

67.149 

0.9999 

Explanation  on  page  39 


66 


Miscellaneous  Tables 


49 

.  Napierian  Logarithms  of  Numbers  from  i  to  119 

n 

o.            I.          2.            3.          4.          5.            6.            7.          8.          g. 

o 

—  00 

0.0000 

0-6931 

I . 0986 

1.3863 

1.6094 

1.7918 

I  - 94S9 

2-0794 

2. 1972 

I 

2.3026 

2-3979 

2  -  4849 

2-5649 

2.6391 

2.7081 

2.7726 

2.8332 

2.8904 

2.9444 

2 

2   9957 

3  044.5 

3-0910 

3-1355 

3-1781 

3-2189 

3-2581 

3-2958 

3-3322 

3   3673 

3 

3.4012 

3-4340 

3-4657 

3-4965 

3- 5264 

3-SSS3 

3-5835 

3.6109 

3   6376 

3-6636 

4 

3.6889 

3-7136 

3-7377 

3.7612 

3-7842 

3-8067 

3-8286 

3.8501 

3-8712 

3.8918 

5 

3.9120 

3-9318 

3-9512 

3   9703 

3 • 9890 

4-0073 

4-0254 

4-0430 

4.0604 

4-0775 

6 

4  094.? 

4-I109 

4-1271 

4-1451 

4-1589 

4-1744 

4. 1S97 

4-2047 

4.2195 

4-2341 

7 

4.2485 

4.2627 

4-2767 

4-2905 

4.3041 

4-3175 

4- 3307 

4- 3438 

4-3567 

4-3694 

8 

4.3820 

4-3944 

4-4067 

4.4188 

4-4308 

4-4427 

4-4543 

4-4659 

4-4773 

4-4886 

Q 

4.4998 

4-5109 

4-5218 

4-5326 

4-5433 

4-5539 

4- 5643 

4-5747 

4- 5850 

4-5951 

10 

4-6052 

4.6151 

4.6250 

4- 6347 

4.6444 

4.6540 

4-6634 

4.6728 

4.6821 

4.6913 

II 

4,7005 

4.7095 

4-7185 

4- 7274 

4-7362 

4-7449 

4-7536 

4.7622 

4-7707 

4-7791 

50.  Multipliers  for  Transferring  Logarithms 


Common  to  Napierian 


2.302585093 
4.605170186 
6-907755279 

9.210340372 
11.512925465 
13.815510558 

16. 118095651 
18.420680744 
20.723265837 


Example. 
Find  Nap  log  of  105 

Com  log  105  =  2.02119 

2  4.605170 

.02  46052 

I  2303 

I  230 

9  207 


Nap  log  105  =4.65396 


Napierian  to  Common 


0.434294482 
0.868588964 
I . 302883446 

I -737177928 

2- I714724IO 
2.605766891 

3.040061373 
3.474355855 
3.908650337 


E.xample. 

Find  number  correspond 
ing  to  Nap  log  1.6078 


.6 


07 


0.26058 

304 

35 


+0.26397 
-0.43429 


Com  log  = 


1.8297 


Number  =  0.6756 


61.  Explanations 

Table  46  gives  seven-place  constants  and  their  logarithms 
which  will  often  be  of  use  in  niathematical  computations;  tt  is  the 
ratio  of  the  circumference  of  a  circle  to  its  diameter,  e  is  the  base 
of  the  Napierian  system  of  logarithms,  and  {x  is  the  modulus  of  the 
common  system  of  logarithms.  In  this  table  the  shilling  mark  / 
denotes  division;  thus,  tt/SO  is  the  same  as  ^V 't. 

Table  47  gives  decimal  equivalents  of  some  common  frac- 
tions, thus,  7/32=0.21875.  The  logarithms  may  be  u.«!eful  in 
computations;  thus  to  divide  0.3275  by  13/16  the  log  of  13/16 
is  subtracted  from  the  log  of  0.3275,  or  1.51521  -I.909S2  =  1.60539 
which  is  the  log  of  0.4031. 


Miscellaneous  Tables  '    '   ^ '    '  ^ '  gy 

Table  48  gives  natural  hyperbolic  sines,  cosines  and  tangents  of 
numbers.  These  are  useful  in  engineering  problems  relating  to 
beams,  to  the  catenary,  the  parabola,  and  other  curves,  also  in 
the  theory  of  alternating  currents.  Hyperbolic  functions  can  be 
graphically  represented  in  a  rectangular  hyperbola  in  the  same 
way  as  trigonometric  functions  are  represented  in  a  circle.  "  Sinh" 
is  the  abbreviation  for  the  hyperbolic  sine,  and  "  cosh  "  for  the 
hyperbolic  cosine;  sinh  is  usually  pronounced  shin. 

Table  49  gives  a  few  Napierian  logarithms,  often  called  hyper- 
bolic logarithms.  The  base  of  this  system  is  the  number  2.71828. 
Such  logarithms  arise  in  many  formulas  deduced  by  calculus, 
and  they  are  widely  used  in  physical  and  engineering  problems. 
Table  50  shows  how  to  obtain  the  Napierian  logarithm  of  any 
number  from  its  common  logarithm. 

51.  Exercises 

I.  Divide  4738  by  ir,  using  logarithms. 
2    What  is  the  value  of  Air^? 

3.  What  is  the  meaning  of  ISOVtt? 

4.  What  is  the  area  of  a  sphere  whose  radius  is  100  feet? 

5.  What  is  the  volume  of  a  sphere  whose  diameter  is  10  centi- 
meters? 

6.  Find  the  value  of  472/V|'. 

7.  What  are  the  decimal  equivalents  of  2/7,  11/32,  80/32,  7/12 
9/12,  34/64,  and  35/64? 

8.  Given  m  =  1.25;  square  sinh  u  and  cosh  u  and  subtract  the  first 
square  from  the  second.     Also  do  the  same  for  another  value  of  u. 

9.  Is  tanh  u  equal  to  sinh  t</cosh  u? 

10.  Multiply  37  by  8,  using  Napierian  logarithms. 

II.  Divide  119  by  17,  using  Napierian  logarithms. 

12.  Taking  the  common  logarithm  of  e  from  Table  47,  find  the 
value  of  e^ 

13.  If  the  common  logarithm  of  a  number  is  4.0000,  what  i."  its 
Napierian  logarithm? 

14.  Find  the  Napierian  logarithm  of  2.71828^ 

15.  Find  the  Napierian  logarithms  of  3275,  3.275,  1800,  and  0.18 


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